Subalgebras of the rank two semisimple Lie algebras

Andrew Douglas; Joe Repka

arXiv:1705.03600·math.RA·August 21, 2024

Subalgebras of the rank two semisimple Lie algebras

Andrew Douglas, Joe Repka

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TL;DR

This paper provides a comprehensive classification of subalgebras within rank 2 semisimple Lie algebras, including classical and exceptional types, using a unified scheme over complex numbers.

Contribution

It completes the classification of subalgebras for all rank 2 semisimple Lie algebras, including the exceptional G2, with a uniform approach.

Findings

01

Classification of subalgebras of classical rank 2 Lie algebras

02

Completion of G2 subalgebra classification by Mayanskiy

03

Unified scheme for low-dimensional Lie algebra classification

Abstract

In this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the subalgebras of the classical rank 2 semisimple Lie algebras. Finally, Mayanskiy finished the classification of the subalgebras of the remaining rank 2 semisimple Lie algebra, the exceptional Lie algebra $G_{2}$ . We identify subalgebras of the classification in terms of a uniform classification scheme of Lie algebras of low dimension. The classification is up to inner automorphism, and the ground field is the complex numbers.

Figures4

Click any figure to enlarge with its caption.

Tables22

Table 1. Table 1. One-dimensional (solvable) subalgebras of C 2 subscript 𝐶 2 C_{2}

Representative	Conditions	Equivalences	Isomorphism Class
Semisimple
$⟨ T_{a, b} ⟩$	$a, b \neq 0$	$⟨ T_{a, b} ⟩ \sim ⟨ T_{a^{'}, b^{'}} ⟩$	$𝔫_{1, 1}$
	$a \neq \pm b$	iff ${a, b} = {λ a^{'}, \pm λ b^{'}}$
		for some $λ \in ℂ^{*}$
$⟨ T_{1, 0} ⟩$			$𝔫_{1, 1}$
$⟨ T_{1, 1} ⟩$			$𝔫_{1, 1}$
Nilpotent
$⟨ X_{α} ⟩$			$𝔫_{1, 1}$
$⟨ X_{β} ⟩$			$𝔫_{1, 1}$
$⟨ X_{α} + X_{β} ⟩$			$𝔫_{1, 1}$
Non-Trivial Jordan Decomposition
$⟨ T_{1, 0} + X_{α} ⟩$			$𝔫_{1, 1}$
$⟨ T_{1, 1} + X_{β} ⟩$			$𝔫_{1, 1}$

Table 2. Table 2. Two-dimensional (solvable) subalgebras of C 2 subscript 𝐶 2 C_{2}

Representative	Conditions	Equivalences	Isomorphism Class
Semisimple
$𝔥$			2 $𝔫_{1, 1}$
Containing both Semisimple and Nilpotent Elements
$⟨ T_{3, 1}, X_{α} + X_{β} ⟩$			$𝔰_{2, 1}$
$⟨ T_{a, 1}, X_{α} ⟩$	$a \neq 0, \pm 1$	$⟨ T_{a, 1}, X_{α} ⟩ \sim ⟨ T_{- a, 1}, X_{α} ⟩$	$𝔰_{2, 1}$
$⟨ T_{a, 1}, X_{β} ⟩$	$a \neq 0, \pm 1$	$⟨ T_{a, 1}, X_{β} ⟩ \sim ⟨ T_{a^{- 1}, 1}, X_{β} ⟩$	$𝔰_{2, 1}$
$⟨ T_{1, 0}, X_{α} ⟩$			2 $𝔫_{1, 1}$
$⟨ T_{1, 0}, X_{β} ⟩$			$𝔰_{2, 1}$
$⟨ T_{1, 0}, X_{α + 2 β} ⟩$			$𝔰_{2, 1}$
$⟨ T_{1, 1}, X_{α} ⟩$			$𝔰_{2, 1}$
$⟨ T_{1, 1}, X_{β} ⟩$			2 $𝔫_{1, 1}$
$⟨ T_{1, 1}, X_{α + β} ⟩$			$𝔰_{2, 1}$
Containing no Semisimple Elements, but not Nilpotent
$⟨ T_{1, 1} + X_{β}, X_{α + 2 β} ⟩$			$𝔰_{2, 1}$
$⟨ T_{1, 0} + X_{α}, X_{α + β} ⟩$			$𝔰_{2, 1}$
$⟨ T_{1, 0} + X_{α}, X_{α + 2 β} ⟩$			$𝔰_{2, 1}$
Nilpotent
$⟨ X_{α}, X_{α + β} ⟩$			2 $𝔫_{1, 1}$
$⟨ X_{α}, X_{α + 2 β} ⟩$			2 $𝔫_{1, 1}$
$⟨ X_{β} + X_{α}, X_{α + 2 β} ⟩$			2 $𝔫_{1, 1}$

Table 3. Table 3. Three-dimensional solvable subalgebras of C 2 subscript 𝐶 2 C_{2}

Representative	Conditions	Equivalences	Isomorphism Class
Containing a Cartan Subalgebra
$⟨ 𝔥, X_{α} ⟩$			$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$⟨ 𝔥, X_{β} ⟩$			$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
Containing a Regular Semisimple Element but not a Cartan Subalgebra
$⟨ T_{a, 1}, X_{α}, X_{α + β} ⟩$	$a \neq 0, \pm 1, - 3$		$𝔰_{3, 1, A = \frac{1 + 2 α + \sqrt{1 + 4 α}}{- 2 α}}$
			where $α = - 2 (a + 1) / {(a + 3)}^{2}$
$⟨ T_{- 3, 1}, X_{α}, X_{α + β} ⟩$			$𝔰_{3, 1, A = - 1}$
$⟨ T_{a, 1}, X_{α}, X_{α + 2 β} ⟩$	$a \neq 0, \pm 1$	$⟨ T_{a, 1}, X_{α}, X_{α + 2 β} ⟩ \sim ⟨ T_{a^{- 1}, 1}, X_{α}, X_{α + 2 β} ⟩$	$𝔰_{3, 1, A = \frac{1 + 2 α + \sqrt{1 + 4 α}}{- 2 α}}$
			where $α = - a / {(a + 1)}^{2}$
$⟨ T_{3, 1}, X_{α} + X_{β}, X_{α + 2 β} ⟩$			$𝔰_{3, 1, A = 1 / 3}$
Containing a Non-regular Semisimple Element but not a Cartan Subalgebra
$⟨ T_{1, 0}, X_{α}, X_{α + β} ⟩$			$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$⟨ T_{1, 0}, X_{α}, X_{α + 2 β} ⟩$			$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$⟨ T_{1, 0}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{3, 1, A = 1 / 2}$
$⟨ T_{1, - 1}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$⟨ T_{1, - 1}, X_{α}, X_{α + 2 β} ⟩$			$𝔰_{3, 1, A = - 1}$
$⟨ T_{1, - 1}, X_{β}, X_{α + 2 β} ⟩$			$𝔰_{3, 1, A = 1}$
$⟨ T_{1, 1}, X_{α}, X_{α + 2 β} ⟩$			$𝔰_{3, 1, A = 1}$
Containing no Semisimple Elements but not Nilpotent
$⟨ T_{1, 1} + X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{3, 2}$
$⟨ T_{1, - 1} + X_{α + β}, X_{α}, X_{α + 2 β} ⟩$			$𝔰_{3, 1, A = - 1}$
$⟨ T_{1, 0} + X_{α}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{3, 1, A = 1 / 2}$
Nilpotent
$⟨ X_{α}, X_{α + β}, X_{α + 2 β} ⟩$			$3 𝔫_{1, 1}$
$⟨ X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔫_{3, 1}$
$⟨ X_{α} + X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔫_{3, 1}$

Table 4. Table 4. Four-dimensional solvable subalgebras of C 2 subscript 𝐶 2 C_{2}

Representative	Conditions	Equivalences	Isomorphism Class
Containing a Cartan Subalgebra
$⟨ 𝔥, X_{α}, X_{α + β} ⟩$			$𝔰_{4, 12}$
$⟨ 𝔥, X_{α}, X_{α + 2 β} ⟩$			$𝔰_{4, 12}$
Containing a Regular Semisimple Element but not a Cartan Subalgebra
$⟨ T_{a, 1}, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$	$a \neq 0, \pm 1$	$⟨ T_{a, 1}, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$	$𝔰_{4, 2}$ , $𝔰_{4, 3}$ , $𝔰_{4, 4}$ ,
		$\sim ⟨ T_{- a, 1}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{3, 1}$ , or $𝔫_{1, 1} \oplus 𝔰_{3, 2}$
		$\sim ⟨ T_{a^{- 1}, 1}, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$	(depending on $a$ , c.f., [DR17])
$⟨ T_{a, 1}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$	$a \neq 0, \pm 1$	$⟨ T_{a, 1}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$	$𝔰_{4, 8, A = \frac{1 + 2 α + \sqrt{1 + 4 α}}{- 2 α}}$
		$\sim ⟨ T_{- a, 1}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$	where $α = (1 - a^{2}) / (4 a^{2})$
$⟨ T_{3, 1}, X_{α} + X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{4, 8, A = 1 / 2}$
Containing a Non-regular Semisimple Element but not a Cartan Subalgebra
$⟨ T_{0, 1}, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔫_{1, 1} \oplus 𝔰_{3, 1, A = 1 / 2}$
$⟨ T_{0, 1}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{4, 6}$
$⟨ T_{1, 0}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{4, 8, A = 1}$
$⟨ T_{1, 1}, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{4, 3, A = B = 1}$
$⟨ T_{1, - 1}, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔫_{1, 1} \oplus 𝔰_{3, 1, A = - 1}$
$⟨ T_{1, 1}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{4, 11}$
Containing no Semisimple Elements but not Nilpotent
$⟨ T_{1, 1} + X_{β}, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{4, 2}$
$⟨ T_{1, 0} + X_{α}, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$			$𝔰_{4, 10}$
Nilpotent Subalgebras
$𝔫$			$𝔫_{4, 1}$

Table 5. Table 5. Five- and six-dimensional solvable subalgebras of C 2 subscript 𝐶 2 C_{2}

Dimension	Representative	Conditions	Isomorphism Class
Containing a Cartan Subalgebra
$5$	$⟨ 𝔥, X_{α}, X_{α + β}, X_{α + 2 β} ⟩$		$𝔰_{5, 41, A = B = \frac{1}{2}}$
$5$	$⟨ 𝔥, X_{β}, X_{α + β}, X_{α + 2 β} ⟩$		$𝔰_{5, 44}$
Containing a Regular Semisimple Element but not a Cartan Subalgebra
$5$	$⟨ T_{a, 1}, 𝔫 ⟩$	$a \neq 0, \pm 1$	$𝔰_{5, 35, A = \frac{2}{a - 1}}$
Containing a Non-regular Semisimple Element but not a Cartan Subalgebra
$5$	$⟨ T_{1, - 1}, 𝔫 ⟩$		$𝔰_{5, 35, A = - 1}$
$5$	$⟨ T_{1, 1}, 𝔫 ⟩$		$𝔰_{5, 37}$
$5$	$⟨ T_{1, 0}, 𝔫 ⟩$		$𝔰_{5, 36}$
$5$	$⟨ T_{0, 1}, 𝔫 ⟩$		$𝔰_{5, 33}$
$6$	$𝔟$		$𝔰_{6, 242}$

Table 6. Table 6. Semisimple and Levi decomposable subalgebras of C 2 subscript 𝐶 2 C_{2}

Dimension	Representative	Isomorphism Class
Semisimple Subalgebras
$3$	$⟨ X_{α + 2 β}, Y_{α + 2 β}, T_{1, 0} ⟩$	$A_{1}$
$3$	$⟨ X_{α + β}, Y_{α + β}, T_{1, 1} ⟩$	$A_{1}$
$3$	$⟨ X_{α} + X_{β}, Y_{α} + Y_{β}, T_{3, 1} ⟩$	$A_{1}$
$6$	$⟨ X_{α}, Y_{α}, T_{0, 1} ⟩ \oplus ⟨ X_{α + 2 β}, Y_{α + 2 β}, T_{1, 0} ⟩$	$A_{1} \times A_{1}$
Levi Decomposable Subalgebras
$4$	$⟨ X_{α + β}, Y_{α + β}, T_{1, 1} ⟩ \oplus ⟨ T_{1, - 1} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ X_{α + 2 β}, Y_{α + 2 β}, T_{1, 0} ⟩ \oplus ⟨ X_{α} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ X_{α + 2 β}, Y_{α + 2 β}, T_{1, 0} ⟩ \oplus ⟨ T_{0, 1} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$5$	$⟨ X_{α + 2 β}, Y_{α + 2 β}, T_{1, 0} ⟩ \oplus ⟨ X_{α}, T_{0, 1} ⟩$	$A_{1} \oplus 𝔰_{2, 1}$
$6$	$⟨ X_{α + β}, Y_{α + β}, T_{1, 1} ⟩ ⨭ ⟨ X_{α}, Y_{β}, Y_{α + 2 β} ⟩$	$A_{1} ⨭ 3 𝔫_{1, 1}$
$6$	$⟨ X_{α + 2 β}, Y_{α + 2 β}, T_{1, 0} ⟩ ⨭ ⟨ X_{β}, Y_{α + β}, Y_{α} ⟩$	$A_{1} ⨭ 𝔫_{3, 1}$
$7$	$⟨ X_{α + β}, Y_{α + β}, T_{1, 1} ⟩ ⨭ ⟨ X_{α}, Y_{β}, Y_{α + 2 β}, T_{1, - 1} ⟩$	$A_{1} ⨭ 𝔰_{4, 3, A = B = 1}$
$7$	$⟨ X_{α + 2 β}, Y_{α + 2 β}, T_{1, 0} ⟩ ⨭ ⟨ X_{β}, Y_{α + β}, Y_{α}, T_{0, 1} ⟩$	$A_{1} ⨭ 𝔰_{4, 8, A = 1}$

Table 7. Table 7. One- and two-dimensional (solvable) subalgebras of A 2 subscript 𝐴 2 A_{2}

Dimension	Representative	Isomorphism Class
$1$	$⟨ X_{α} + X_{β} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ X_{α} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ X_{α} + H_{α} + 2 H_{β} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ H_{α} + a H_{β} ⟩$	$𝔫_{1, 1}$
	$⟨ H_{α} + a H_{β} ⟩ \sim ⟨ H_{α} + b H_{β} ⟩$
	iff $b = a, \frac{1}{a}, 1 - a, \frac{1}{1 - a}, \frac{a}{a - 1},$ or $\frac{a - 1}{a}$
$2$	$⟨ X_{α} + X_{β}, X_{α + β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ X_{α}, H_{α} + 2 H_{β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ X_{α}, X_{α + β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ X_{α}, Y_{β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ H_{α}, H_{β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ X_{α} + X_{β}, H_{α} + H_{β} ⟩$	$𝔰_{2, 1}$
$2$	$⟨ X_{α}, - H_{α} + H_{β} + 3 X_{α + β} ⟩$	$𝔰_{2, 1}$
$2$	$⟨ X_{α}, - 2 H_{α} - H_{β} + 3 Y_{β} ⟩$	$𝔰_{2, 1}$
$2$	$⟨ X_{α}, a H_{α} + (2 a + 1) H_{β} ⟩$	$𝔰_{2, 1}$
	$⟨ X_{α}, a H_{α} + (2 a + 1) H_{β} ⟩ \sim ⟨ X_{α}, b H_{α} + (2 b + 1) H_{β} ⟩$
	iff $a = b$

Table 8. Table 8. Three-, four-, and five-dimensional solvable subalgebras of A 2 subscript 𝐴 2 A_{2}

Dimension	Representative	Isomorphism Class	Conditions
$3$	$⟨ X_{α}, X_{α + β}, 2 H_{α} + H_{β} ⟩$	$𝔰_{3, 1, A = 1}$
$3$	$⟨ X_{α}, Y_{β}, H_{α} - H_{β} ⟩$	$𝔰_{3, 1, A = 1}$
$3$	$⟨ X_{α}, X_{α + β}, 2 H_{α} + H_{β} + X_{β} ⟩$	$𝔰_{3, 2}$
$3$	$⟨ Y_{α}, Y_{α + β}, 2 H_{α} + H_{β} + X_{β} ⟩$	$𝔰_{3, 2}$
$3$	$⟨ X_{α} + X_{β}, X_{α + β}, H_{α} + H_{β} ⟩$	$𝔰_{3, 1, A = 1 / 2}$
$3$	$⟨ X_{α}, H_{α}, H_{β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$3$	$⟨ X_{α}, X_{α + β}, (a - 1) H_{α} + a H_{β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$	if $a = \frac{1}{2}, 2$
	$a \neq \pm 1$	$𝔰_{3, 1, \frac{1 + 2 α + \sqrt{1 + 4 α}}{- 2 a α}}$ ,	otherwise
		$α = - \frac{(2 a - 1) (a - 2)}{9 {(a - 1)}^{2}}$
$3$	$⟨ X_{α}, Y_{β}, H_{α} + a H_{β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$	if $a = \frac{1}{2}, 2$
	$a \neq \pm 1$	$𝔰_{3, 1, \frac{1 + 2 α + \sqrt{1 + 4 α}}{- 2 a α}}$ ,	otherwise
		$α = - \frac{(2 a - 1) (a - 2)}{9 {(a - 1)}^{2}}$
$3$	$⟨ X_{α}, X_{α + β}, H_{β} ⟩$	$𝔰_{3, 1, A = - 1}$
$3$	$⟨ X_{α}, Y_{β}, H_{α} + H_{β} ⟩$	$𝔰_{3, 1, A = - 1}$
$3$	$⟨ X_{α}, X_{β}, X_{α + β} ⟩$	$𝔫_{3, 1}$
$4$	$⟨ X_{α}, X_{α + β}, H_{α}, H_{β} ⟩$	$𝔰_{4, 12}$
$4$	$⟨ X_{α}, Y_{β}, H_{α}, H_{β} ⟩$	$𝔰_{4, 12}$
$4$	$⟨ X_{α}, X_{β}, X_{α + β}, H_{α} + H_{β} ⟩$	$𝔰_{4, 8, A = 1}$
$4$	$⟨ X_{α}, X_{β}, X_{α + β}, a H_{α} + H_{β} ⟩$	$𝔰_{4, 11}$	if $a = \frac{1}{2}, 2$
	$a \neq \pm 1$
	$⟨ X_{α}, X_{β}, X_{α + β}, a H_{α} + H_{β} ⟩ \sim ⟨ X_{α}, X_{β}, X_{α + β}, b H_{α} + H_{β} ⟩$	$𝔰_{4, 8, \frac{1 + 2 α + \sqrt{1 + 4 α}}{- 2 α}}$ ,	otherwise
	iff $a = b$	$α = \frac{(2 a - 1) (a - 2)}{{(a + 1)}^{2}}$
$4$	$⟨ X_{β}, Y_{α}, Y_{α + β}, 2 H_{α} + H_{β} ⟩$	$𝔰_{4, 11}$
$4$	$⟨ X_{α}, X_{β}, X_{α + β}, H_{α} ⟩$	$𝔰_{4, 8, A = - 2}$
$4$	$⟨ X_{α}, X_{β}, X_{α + β}, H_{α} - H_{β} ⟩$	$𝔰_{4, 6}$
$5$	$⟨ X_{α}, X_{β}, X_{α + β}, H_{α}, H_{β} ⟩$	$𝔰_{5, 44}$

Table 9. Table 9. Semisimple and Levi decomposable subalgebras of A 2 subscript 𝐴 2 A_{2}

Dimension	Representative	Isomorphism Class
Semisimple Subalgebras
$3$	$⟨ X_{α + β}, Y_{α + β}, H_{α} + H_{β} ⟩$	$A_{1}$
$3$	$⟨ X_{α} + X_{β}, 2 Y_{α} + 2 Y_{β}, 2 H_{α} + 2 H_{β} ⟩$	$A_{1}$
Levi Decomposable Subalgebras
$4$	$⟨ X_{α + β}, Y_{α + β}, H_{α} + H_{β} ⟩ \oplus ⟨ H_{α} - H_{β} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$5$	$⟨ X_{α + β}, Y_{α + β}, H_{α} + H_{β} ⟩ ⨭ ⟨ X_{α}, Y_{β} ⟩$	$A_{1} ⨭ 2 𝔫_{1, 1}$
$5$	$⟨ X_{α + β}, Y_{α + β}, H_{α} + H_{β} ⟩ ⨭ ⟨ X_{β}, Y_{α} ⟩$	$A_{1} ⨭ 2 𝔫_{1, 1}$
$6$	$⟨ X_{α + β}, Y_{α + β}, H_{α} + H_{β} ⟩ ⨭ ⟨ X_{α}, Y_{β}, H_{α} - H_{β} ⟩$	$A_{1} ⨭ 𝔰_{3, 1, A = 1}$
$6$	$⟨ X_{α + β}, Y_{α + β}, H_{α} + H_{β} ⟩ ⨭ ⟨ X_{β}, Y_{α}, H_{α} - H_{β} ⟩$	$A_{1} ⨭ 𝔰_{3, 1, A = 1}$

Table 10. Table 10. Solvable subalgebras of A 1 × A 1 subscript 𝐴 1 subscript 𝐴 1 A_{1}\times A_{1}

Dimension	Representative	Isomorphism Class
$1$	$⟨ X_{α} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ X_{β} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ H_{α} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ H_{β} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ X_{α} + X_{β} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ X_{α} + H_{β} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ H_{α} + X_{β} ⟩$	$𝔫_{1, 1}$
$1$	$⟨ H_{α} + a H_{β} ⟩$ , $a \in ℂ^{*}$	$𝔫_{1, 1}$
	$⟨ H_{α} + a H_{β} ⟩ \sim ⟨ H_{α} + b H_{β} ⟩$ iff $a = \pm b$
$2$	$⟨ X_{α}, X_{β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ X_{α}, H_{β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ H_{α}, X_{β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ H_{α}, H_{β} ⟩$	$2 𝔫_{1, 1}$
$2$	$⟨ X_{α} + X_{β}, H_{α} + H_{β} ⟩$	$𝔰_{2, 1}$
$2$	$⟨ X_{α}, H_{α} + a H_{β} ⟩$ , $a \in ℂ$	$𝔰_{2, 1}$
	$⟨ X_{α}, H_{α} + a H_{β} ⟩ \sim ⟨ X_{α}, H_{α} + b H_{β} ⟩$ iff $a = \pm b$
$2$	$⟨ X_{α}, H_{α} + X_{β} ⟩$	$𝔰_{2, 1}$
$2$	$⟨ X_{β}, H_{β} + a H_{α} ⟩$ , $a \in ℂ$	$𝔰_{2, 1}$
	$⟨ X_{β}, H_{β} + a H_{α} ⟩ \sim ⟨ X_{β}, H_{β} + b H_{α} ⟩$ iff $a = \pm b$
$2$	$⟨ X_{β}, H_{β} + X_{α} ⟩$	$𝔰_{2, 1}$
$3$	$⟨ X_{α}, X_{β}, H_{α} + H_{β} ⟩$	$𝔰_{3, 1, A = 1}$
$3$	$⟨ X_{α}, X_{β}, (1 + \sqrt{1 + 4 a}) H_{α} + (1 - \sqrt{1 + 4 a}) H_{β} ⟩$	$𝔰_{3, 1, \frac{1 + 2 a + \sqrt{1 + 4 a}}{- 2 a}}$
	$a \in ℂ ∖ {- \frac{1}{4}, 0}$
$3$	$⟨ X_{α}, X_{β}, (1 - \sqrt{1 + 4 a}) H_{α} + (1 + \sqrt{1 + 4 a}) H_{β} ⟩$	$𝔰_{3, 1, \frac{1 + 2 a + \sqrt{1 + 4 a}}{- 2 a}}$
	$a \in ℂ ∖ {- \frac{1}{4}, 0}$
$3$	$⟨ X_{β}, H_{α}, H_{β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$3$	$⟨ X_{α}, X_{β}, H_{β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$3$	$⟨ X_{α}, H_{α}, H_{β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$3$	$⟨ X_{α}, X_{β}, H_{α} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$3$	$⟨ X_{α}, X_{β}, H_{α} - H_{β} ⟩$	$𝔰_{3, 1, A = - 1}$
$4$	$⟨ X_{α}, X_{β}, H_{α}, H_{β} ⟩$	$𝔰_{4, 12}$

Table 11. Table 11. Semisimple and Levi decomposable subalgebras of A 1 × A 1 subscript 𝐴 1 subscript 𝐴 1 A_{1}\times A_{1}

Dimension	Representative	Isomorphism Class
Semisimple Subalgebras
$3$	$⟨ X_{α}, Y_{α}, H_{α} ⟩$	$A_{1}$
$3$	$⟨ X_{β}, Y_{β}, H_{β} ⟩$	$A_{1}$
$3$	$⟨ X_{α} + X_{β}, Y_{α} + Y_{β}, H_{α} + H_{β} ⟩$	$A_{1}$
Levi Decomposable Subalgebras
$4$	$⟨ X_{α}, Y_{α}, H_{α} ⟩ \oplus ⟨ H_{β} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ X_{α}, Y_{α}, H_{α} ⟩ \oplus ⟨ X_{β} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ X_{β}, Y_{β}, H_{β} ⟩ \oplus ⟨ H_{α} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ X_{β}, Y_{β}, H_{β} ⟩ \oplus ⟨ X_{α} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$5$	$⟨ X_{α}, Y_{α}, H_{α} ⟩ \oplus ⟨ X_{β}, H_{β} ⟩$	$A_{1} \oplus 𝔰_{2, 1}$
$5$	$⟨ X_{β}, Y_{β}, H_{β} ⟩ \oplus ⟨ X_{α}, H_{α} ⟩$	$A_{1} \oplus 𝔰_{2, 1}$

Table 12. Table 12. One-dimensional (solvable) subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class
Regular Subalgebras
$⟨ X_{α} ⟩$	$𝔫_{1, 1}$
$⟨ X_{β} ⟩$	$𝔫_{1, 1}$
$⟨ H_{c α + d β} ⟩$	$𝔫_{1, 1}$
$(2 c - 3 d) (- c + 2 d) \geq 0, (c, d) \neq (0, 0)$
Non-Regular Subalgebras
$⟨ X_{α} + X_{3 α + 2 β} ⟩$	$𝔫_{1, 1}$
$⟨ X_{α} + X_{β} ⟩$	$𝔫_{1, 1}$
$⟨ H_{3 α + 2 β} + X_{α} ⟩$	$𝔫_{1, 1}$
$⟨ H_{2 α + β} + X_{β} ⟩$	$𝔫_{1, 1}$

Table 13. Table 13. Two-dimensional (solvable) subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class	Conditions
Regular Subalgebras
$𝔥$	$2 𝔫_{1, 1}$
$⟨ H_{c α + d β}, X_{α} ⟩$	$2 𝔫_{1, 1}$	if $2 c = 3 d$
$(2 c - 3 d) d \geq 0$ , $(c, d) \neq (0, 0)$	$𝔰_{2, 1}$	otherwise
$⟨ H_{c α + d β}, X_{β} ⟩$	$2 𝔫_{1, 1}$	if $c = 2 d$
$(2 d - c) c \geq 0$ , $(c, d) \neq (0, 0)$	$𝔰_{2, 1}$	otherwise
$⟨ X_{β}, X_{3 α + 2 β} ⟩$	$2 𝔫_{1, 1}$
$⟨ X_{α}, X_{3 α + β} ⟩$	$2 𝔫_{1, 1}$
$⟨ X_{α}, X_{3 α + 2 β} ⟩$	$2 𝔫_{1, 1}$
Non-Regular Subalgebras
$⟨ X_{α} + X_{β}, X_{3 α + 2 β} ⟩$	$2 𝔫_{1, 1}$
$⟨ X_{β} + X_{3 α + β}, X_{2 α + β} ⟩$	$2 𝔫_{1, 1}$
$⟨ X_{α + β} + X_{3 α + β}, X_{3 α + 2 β} ⟩$	$2 𝔫_{1, 1}$
$⟨ H_{3 α + β}, X_{α} + X_{3 α + 2 β} ⟩$	$𝔰_{2, 1}$
$⟨ H_{9 α + 5 β}, X_{α} + X_{β} ⟩$	$𝔰_{2, 1}$
$⟨ H_{3 α + 2 β} + X_{α}, X_{3 α + β} ⟩$	$𝔰_{2, 1}$
$⟨ H_{3 α + 2 β} + X_{α}, X_{3 α + 2 β} ⟩$	$𝔰_{2, 1}$
$⟨ H_{2 α + β} + X_{β}, X_{α + β} ⟩$	$𝔰_{2, 1}$
$⟨ H_{2 α + β} + X_{β}, X_{2 α + β} ⟩$	$𝔰_{2, 1}$
$⟨ H_{2 α + β} + X_{β}, X_{3 α + 2 β} ⟩$	$𝔰_{2, 1}$

Table 14. Table 14. Regular three-dimensional solvable subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class	Conditions
$⟨ H_{3 α + 2 β}, H_{α}, X_{α} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$⟨ H_{2 α + β}, H_{β}, X_{β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$
$⟨ H_{c α + d β}, X_{α}, X_{3 α + β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$	if $2 c = 3 d$ or $c = d$
	$𝔰_{3, 1, A}$	otherwise
		$A = \frac{2 c - 3 d}{3 (c - d)}$ , or $\frac{3 (c - d)}{2 c - 3 d}$
		$\| A \| = \min {\| \frac{2 c - 3 d}{3 (c - d)} \|, \| \frac{3 (c - d)}{2 c - 3 d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
$⟨ H_{c α + d β}, X_{α}, X_{3 α + 2 β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$	if $2 c = 3 d$ or $d = 0$
	$𝔰_{3, 1, A}$	otherwise
		$A = \frac{2 c - 3 d}{3 d}$ , or $\frac{3 d}{2 c - 3 d}$
		$\| A \| = \min {\| \frac{2 c - 3 d}{3 d} \|, \| \frac{3 d}{2 c - 3 d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
$⟨ H_{c α + d β}, X_{β}, X_{3 α + 2 β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{2, 1}$	if $c = 2 d$
$(3 d - c) (c - d) \geq 0, (c, d) \neq (0, 0)$	$𝔰_{3, 1, A}$	otherwise
		$A = \frac{c - 2 d}{d}$ , or $\frac{d}{c - 2 d}$
		$\| A \| = \min {\| \frac{c - 2 d}{d} \|, \| \frac{d}{c - 2 d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
$⟨ X_{α}, X_{2 α + β}, X_{3 α + β} ⟩$	$𝔫_{3, 1}$
$⟨ X_{β}, X_{α + β}, X_{3 α + 2 β} ⟩$	$3 𝔫_{1, 1}$
$⟨ X_{β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{3, 1}$
$⟨ X_{α}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$3 𝔫_{1, 1}$

Table 15. Table 15. Non-regular three-dimensional solvable subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class
$⟨ X_{α} + X_{β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{3, 1}$
$⟨ X_{β} + X_{3 α + β}, X_{2 α + β}, X_{3 α + 2 β} ⟩$	$3 𝔫_{1, 1}$
$⟨ X_{α + β} + X_{3 α + β}, X_{2 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{3, 1}$
$⟨ X_{β} + X_{α + β} + λ X_{3 α + β}, X_{2 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{3, 1}$
$λ \in ℂ^{*}$
$⟨ H_{9 α + 5 β}, X_{α} + X_{β}, X_{3 α + 2 β} ⟩$	$𝔰_{3, 1, A = 1 / 5}$
$⟨ H_{3 α + 2 β}, X_{β} + X_{3 α + β}, X_{2 α + β} ⟩$	$𝔰_{3, 1, A = 1}$
$⟨ H_{3 α + 2 β}, X_{α + β} + X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{3, 1, A = 1 / 2}$
$⟨ H_{α} + X_{3 α + 2 β}, X_{α}, X_{3 α + β} ⟩$	$𝔰_{3, 1, A = 2 / 3}$
$⟨ H_{α} + X_{3 α + 2 β}, X_{β}, X_{α + β} ⟩$	$𝔰_{3, 1, A = 1 / 3}$
$⟨ H_{α + β} + X_{3 α + β}, X_{α}, X_{3 α + 2 β} ⟩$	$𝔰_{3, 1, A = - 1 / 3}$
$⟨ H_{α + β} + X_{3 α + β}, X_{β}, X_{3 α + 2 β} ⟩$	$𝔰_{3, 2}$
$⟨ H_{β} + X_{2 α + β}, X_{α}, X_{3 α + β} ⟩$	$𝔰_{3, 2}$
$⟨ H_{β} + X_{2 α + β}, X_{β}, X_{3 α + 2 β} ⟩$	$𝔰_{3, 1, A = 1 / 2}$
$⟨ H_{3 α + β} + X_{α + β}, X_{β}, X_{3 α + 2 β} ⟩$	$𝔰_{3, 1, A = - 1}$

Table 16. Table 16. Regular four-dimensional solvable subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class	Conditions
$⟨ H_{α}, H_{β}, X_{β}, X_{3 α + 2 β} ⟩$	$2 𝔰_{2, 1}$
$⟨ H_{α}, H_{β}, X_{α}, X_{3 α + β} ⟩$	$2 𝔰_{2, 1}$
$⟨ H_{α}, H_{β}, X_{α}, X_{3 α + 2 β} ⟩$	$2 𝔰_{2, 1}$
$⟨ X_{α}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{3, 1} \oplus 𝔫_{1, 1}$
$⟨ X_{β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{3, 1} \oplus 𝔫_{1, 1}$
$⟨ H_{c α + d β}, X_{α}, X_{2 α + β}, X_{3 α + β} ⟩$	$𝔰_{4, 11}$	if $c = 0$ or $2 c = 3 d$
	$𝔰_{4, 6}$	if $c = d$
	$𝔰_{4, 8, A}$	otherwise
		$A = \frac{2 c - 3 d}{c}$ or $\frac{c}{2 c - 3 d}$
		$\| A \| = \min {\| \frac{2 c - 3 d}{c} \|, \| \frac{c}{2 c - 3 d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
$⟨ H_{c α + d β}, X_{β}, X_{α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{3, 1, A = 1 / 3}$	if $c = 2 d$ or $d = 0$
	$𝔫_{1, 1} \oplus 𝔰_{3, 1, A = - 1}$	if $c = 3 d$
	$𝔰_{4, 3, A, B}$	otherwise
		$A, B \in {\frac{- c + 3 d}{- 3 c + 3 d}, \frac{d}{- c + d}, \frac{- 3 c + 6 d}{- c + 3 d},$
		$\frac{3 d}{- c + 3 d}, \frac{- c + 2 d}{d}, \frac{- c + 3 d}{3 d}}$
		$0 < \| B \| \leq \| A \| \leq 1$
$⟨ H_{c α + d β}, X_{β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 11}$	if $c = 2 d$
$(2 c - 3 d) d \geq 0, (c, d) \neq (0, 0)$	$𝔰_{4, 6}$	if $d = 0$
	$𝔰_{4, 8, A}$	otherwise
		$A = \frac{c - d}{- c + 2 d}$ or $\frac{- c + 2 d}{c - d}$
		$\| A \| = \min {\| \frac{c - d}{- c + 2 d} \|, \| \frac{- c + 2 d}{c - d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
$⟨ H_{c α + d β}, X_{α}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{1, 1} \oplus 𝔰_{3, 1, A = 1 / 2}$	if $2 c = 3 d$
	$𝔫_{1, 1} \oplus 𝔰_{3, 1, A = - 1 / 3}$	if $c = d$
	$𝔫_{1, 1} \oplus 𝔰_{3, 1, A = 2 / 3}$	if $d = 0$
	$𝔰_{4, 3, A, B}$	otherwise
		$A, B \in {\frac{2 c - 3 d}{3 c - 3 d}, \frac{d}{c - d}, \frac{3 c - 3 d}{2 c - 3 d},$
		$\frac{3 d}{2 c - 3 d}, \frac{c - d}{d}, \frac{2 c - 3 d}{3 d}}$
		$0 < \| B \| \leq \| A \| \leq 1$

Table 17. Table 17. Non-regular four-dimensional solvable subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class
$⟨ X_{α} + X_{β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{4, 1}$
$⟨ X_{β} + X_{2 α + β}, X_{α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{3, 1} \oplus 𝔫_{1, 1}$
$⟨ H_{9 α + 5 β}, X_{α} + X_{β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 8, A = 1 / 4}$
$⟨ H_{3 α + 2 β}, X_{β} + X_{3 α + β}, X_{2 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 3, A = B = 1 / 2}$
$⟨ H_{3 α + 2 β}, X_{α + β} + X_{3 α + β}, X_{2 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 8, A = 1}$
$⟨ H_{3 α + 2 β}, X_{β} + X_{α + β} + λ X_{3 α + β}, X_{2 α + β}, X_{3 α + 2 β} ⟩$ , $λ \in ℂ^{*}$	$𝔰_{4, 8, A = 1}$
$⟨ H_{2 α + β} + X_{β}, X_{α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 4, A = 1 / 3}$
$⟨ H_{3 α + β} + X_{α + β}, X_{β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 8, A = - 1 / 2}$
$⟨ H_{β} + X_{2 α + β}, X_{α}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 4, A = - 1}$
$⟨ H_{β} + X_{2 α + β}, X_{β}, X_{α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 4, A = 2}$
$⟨ H_{α + β} + X_{3 α + β}, X_{β}, X_{α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{4, 4, A = 2 / 3}$
$⟨ H_{α} + X_{3 α + 2 β}, X_{α}, X_{2 α + β}, X_{3 α + β} ⟩$	$𝔰_{4, 8, A = 1 / 2}$

Table 18. Table 18. Five-dimensional solvable subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class	Conditions
Regular Subalgebras
$⟨ H_{α}, H_{β}, X_{α}, X_{2 α + β}, X_{3 α + β} ⟩$	$𝔰_{5, 44}$
$⟨ H_{α}, H_{β}, X_{β}, X_{α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 41, A = B = 1 / 3}$
$⟨ H_{α}, H_{β}, X_{β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 44}$
$⟨ H_{α}, H_{β}, X_{α}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 41, A = 2 / 3, B = - 1 / 3}$
$⟨ H_{c α + d β}, X_{α}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 30, A = 2}$	if $2 c = 3 d$
	$𝔰_{5, 30, A = - 1}$	if $c = 0$
	$𝔰_{5, 17, A = 3}$	if $c = d$
	$𝔫_{1, 1} \oplus 𝔰_{4, 8, A = 1 / 2}$	if $d = 0$
	$𝔰_{5, 22, A, B}$	otherwise
		$A = \frac{2 c - 3 d}{c}$ or $\frac{c}{2 c - 3 d}$
		$\| A \| = \min {\| \frac{2 c - 3 d}{c} \|, \| \frac{c}{2 c - 3 d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
		$B = \frac{3 d}{c}$ or $\frac{3 d}{2 c - 3 d}$
$⟨ H_{c α + d β}, X_{β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 30, A = 2 / 3}$	if $c = 2 d$
	$𝔰_{5, 30, A = 1 / 3}$	if $c = d$
	$𝔰_{5, 17, A = 1 / 3}$	if $d = 0$
	$𝔫_{1, 1} \oplus 𝔰_{4, 8, A = - 1 / 2}$	if $c = 0$
	$𝔰_{5, 22, A, B}$	otherwise
		$A = \frac{c - d}{- c + 2 d}$ or $\frac{- c + 2 d}{c - d}$
		$\| A \| = \min {\| \frac{c - d}{- c + 2 d} \|, \| \frac{- c + 2 d}{c - d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
		$B = \frac{c}{3 c - 3 d}$ or $\frac{c}{- 3 c + 6 d}$
$⟨ X_{α}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{5, 2}$
$⟨ X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{5, 3}$
Non-Regular Subalgebras
$⟨ X_{α} + X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔫_{5, 6}$
$⟨ H_{9 α + 5 β}, X_{α} + X_{β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 35, A = 3}$
$⟨ H_{3 α + 2 β}, X_{β} + X_{2 α + β}, X_{α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 22, A = B = 1}$
$⟨ H_{3 α + β} + X_{α + β}, X_{β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 26, A = - 1 / 2}$
$⟨ H_{3 α + β} + X_{α + β}, X_{α}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 21}$
$⟨ H_{α + β} + X_{3 α + β}, X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{5, 26, A = 1 / 2}$

Table 19. Table 19. Six-dimensional solvable subalgebras of G 2 subscript 𝐺 2 G_{2}

Representative	Isomorphism Class	Conditions
Regular Subalgebras
$⟨ H_{c α + d β}, X_{α}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{6, 150}$	if $c = d$
$(2 d - c) c \geq 0, (c, d) \neq (0, 0)$	$𝔰_{6, 156}$	if $2 c = 3 d$
	$𝔰_{6, 155, A = - 1}$	if $c = 0$
	$𝔰_{6, 155, A}$	otherwise
		$A = \frac{2 c - 3 d}{- c + 3 d}$ or $\frac{- c + 3 d}{2 c - 3 d}$
		$\| A \| = \min {\| \frac{2 c - 3 d}{- c + 3 d} \|, \| \frac{- c + 3 d}{2 c - 3 d} \|}$
		$\arg (A) \leq π$ if $\| A \| = 1$
$⟨ H_{c α + d β}, X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{6, 162, A = 1 / 3}$	if $d = 0$
$(2 c - 3 d) d \geq 0, (c, d) \neq (0, 0)$	$𝔰_{6, 178, A = 1 / 3}$	if $c = 2 d$
	$𝔰_{6, 178, A = 3}$	if $c = 3 d$
	$𝔰_{6, 168, A, B}$	otherwise
		$A, B \in {\pm \frac{- 2 c + 3 d}{d}, \pm \frac{- 2 c + 3 d}{3 d}}$
		$\| B \| \leq \| A \|$ , $\arg (A), \arg (B) < π$
		if $\| A \| = \| B \|$ then $\arg (B) \leq \arg (A)$
$⟨ H_{α}, H_{β}, X_{α}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{6, 234, A = - 1, B = 2}$
$⟨ H_{α}, H_{β}, X_{β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{6, 234, A = 1 / 3, B = 2 / 3}$
$𝔫$	$𝔫_{6, 18}$
Non-Regular Subalgebras
$⟨ H_{9 α + 5 β}, X_{α} + X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{6, 197}$
$⟨ H_{3 α + 2 β} + X_{α}, X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{6, 171}$
$⟨ H_{2 α + β} + X_{β}, X_{α}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{6, 153}$

Table 20. Table 20. Seven- and eight-dimensional solvable subalgebras of G 2 subscript 𝐺 2 G_{2}

Dimension	Representative	Isomorphism Class	Conditions
Regular Subalgebras
$7$	$𝔥 + ⟨ X_{α}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{7, 1}$
$7$	$𝔥 + ⟨ X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$𝔰_{7, 2}$
$7$	$𝔫 + ⟨ H_{c α + d β} ⟩$	$𝔰_{7, 3, c, 3 d}$	$𝔰_{7, 3, c, 3 d} ≅ 𝔰_{7, 3, c^{'}, 3 d^{'}}$
			iff $c d^{'} = c^{'} d$
$8$	$𝔟$	$𝔰_{8, 1}$

Table 21. Table 21. Semisimple subalgebras of G 2 subscript 𝐺 2 G_{2}

Dimension	Representative	Isomorphism Class
Regular Subalgebras
3	$⟨ H_{α}, X_{α}, Y_{α} ⟩$	$A_{1}$
3	$⟨ H_{β}, X_{β}, Y_{β} ⟩$	$A_{1}$
6	$⟨ H_{α}, H_{β}, X_{α}, X_{3 α + 2 β}, Y_{α}, Y_{3 α + 2 β} ⟩$	$A_{1} \times A_{1}$
8	$⟨ H_{α}, H_{β}, X_{β}, X_{3 α + β}, X_{3 α + 2 β}, Y_{β}, Y_{3 α + β}, Y_{3 α + 2 β} ⟩$	$A_{2}$
Non-Regular Subalgebras
3	$⟨ 2 H_{3 α + β}, \sqrt{2} (Y_{β} + X_{3 α + 2 β}), \sqrt{2} (X_{β} + Y_{3 α + 2 β}) ⟩$	$A_{1}$
3	$⟨ 14 H_{9 α + 5 β}, \sqrt{6} X_{α} + \sqrt{10} X_{β}, \sqrt{6} Y_{α} + \sqrt{10} Y_{β} ⟩$	$A_{1}$

Table 22. Table 22. Levi decomposable subalgebras of G 2 subscript 𝐺 2 G_{2}

Dimension	Representative	Isomorphism Class
Regular Subalgebras
$4$	$⟨ H_{α}, X_{α}, Y_{α} ⟩ \oplus ⟨ H_{3 α + 2 β} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ \oplus ⟨ H_{2 α + β} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ H_{α}, X_{α}, Y_{α} ⟩ \oplus ⟨ X_{3 α + 2 β} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$4$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ \oplus ⟨ X_{2 α + β} ⟩$	$A_{1} \oplus 𝔫_{1, 1}$
$5$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ ⨭ ⟨ X_{3 α + β}, X_{3 α + 2 β} ⟩$	$A_{1} ⨭ 2 𝔫_{1, 1}$
$5$	$⟨ H_{α}, X_{α}, Y_{α} ⟩ \oplus ⟨ H_{3 α + 2 β}, X_{3 α + 2 β} ⟩$	$A_{1} \oplus 𝔰_{2, 1}$
$5$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ \oplus ⟨ H_{2 α + β}, X_{2 α + β} ⟩$	$A_{1} \oplus 𝔰_{2, 1}$
$6$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ ⨭ ⟨ H_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$A_{1} ⨭ 𝔰_{3, 1, A = 1}$
$6$	$(⟨ H_{β}, X_{β}, Y_{β} ⟩ ⨭ ⟨ X_{3 α + β}, X_{3 α + 2 β} ⟩) \oplus ⟨ X_{2 α + β} ⟩$	$(A_{1} ⨭ 2 𝔫_{1, 1}) \oplus 𝔫_{1, 1}$
$7$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ ⨭ ⟨ H_{2 α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$A_{1} ⨭ 𝔰_{4, 3, A = 1, B = 2 / 3}$
$8$	$⟨ H_{α}, X_{α}, Y_{α} ⟩ ⨭ ⟨ X_{β}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$A_{1} ⨭ 𝔫_{5, 3}$
$8$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ ⨭ ⟨ X_{α}, X_{α + β}, X_{2 α + β}, X_{3 α + β}, X_{3 α + 2 β} ⟩$	$A_{1} ⨭ 𝔫_{5, 2}$
$9$	$⟨ H_{α}, X_{α}, Y_{α} ⟩ ⨭ ⟨ H_{3 α + 2 β}, X_{β}, X_{α + β}, X_{3 α + 2 β}, X_{2 α + β}, X_{3 α + β} ⟩$	$A_{1} ⨭ 𝔰_{6, 168, A = B = 0}$
$9$	$⟨ H_{β}, X_{β}, Y_{β} ⟩ ⨭ ⟨ H_{2 α + β}, X_{α + β}, X_{3 α + 2 β}, X_{2 α + β}, X_{3 α + β}, X_{α} ⟩$	$A_{1} ⨭ 𝔰_{6, 155, A = 1}$

Equations55

J = 00 - 1 0 000 - 1 10000100 .

J = 00 - 1 0 000 - 1 10000100 .

\bBigg@{3}\{\begin{array}[]{lllllllllll}H_{\alpha},&H_{\beta},\\ X_{\alpha},&X_{\beta},&X_{\alpha+\beta},&X_{\alpha+2\beta},\\ Y_{\alpha},&Y_{\beta},&Y_{\alpha+\beta},&Y_{\alpha+2\beta}\end{array}\bBigg@{3}\},

\bBigg@{3}\{\begin{array}[]{lllllllllll}H_{\alpha},&H_{\beta},\\ X_{\alpha},&X_{\beta},&X_{\alpha+\beta},&X_{\alpha+2\beta},\\ Y_{\alpha},&Y_{\beta},&Y_{\alpha+\beta},&Y_{\alpha+2\beta}\end{array}\bBigg@{3}\},

X_{α} = 0000000000000100,

X_{α} = 0000000000000100,

X_{α + β} = 0000000001001000,

\begin{array}[]{lllllllllllll}\displaystyle Y_{\alpha}&=&X_{\alpha}^{t},&Y_{\beta}&=&X_{\beta}^{t},\\[3.0pt] \displaystyle Y_{\alpha+\beta}&=&X_{\alpha+\beta}^{t},&Y_{\alpha+2\beta}&=&X_{\alpha+2\beta}^{t},\\[3.0pt] \displaystyle H_{\alpha}&=&[X_{\alpha},Y_{\alpha}],&H_{\beta}&=&[X_{\beta},Y_{\beta}].\end{array}

\begin{array}[]{lllllllllllll}\displaystyle Y_{\alpha}&=&X_{\alpha}^{t},&Y_{\beta}&=&X_{\beta}^{t},\\[3.0pt] \displaystyle Y_{\alpha+\beta}&=&X_{\alpha+\beta}^{t},&Y_{\alpha+2\beta}&=&X_{\alpha+2\beta}^{t},\\[3.0pt] \displaystyle H_{\alpha}&=&[X_{\alpha},Y_{\alpha}],&H_{\beta}&=&[X_{\beta},Y_{\beta}].\end{array}

T_{a, b} = (a + b) H_{α} + a H_{β} .

T_{a, b} = (a + b) H_{α} + a H_{β} .

\begin{array}[]{llllllll}\langle X_{\beta}\rangle,&\langle X_{\alpha}\rangle,&\langle X_{\alpha}+X_{\beta}\rangle.\end{array}

\begin{array}[]{llllllll}\langle X_{\beta}\rangle,&\langle X_{\alpha}\rangle,&\langle X_{\alpha}+X_{\beta}\rangle.\end{array}

\begin{array}[]{llllllll}\langle T_{1,b}\rangle\cong\langle T_{1,b^{-1}}\rangle,b\neq 0,\pm 1;&\langle T_{1,0}\rangle;&\langle T_{1,1}\rangle.\end{array}

\begin{array}[]{llllllll}\langle T_{1,b}\rangle\cong\langle T_{1,b^{-1}}\rangle,b\neq 0,\pm 1;&\langle T_{1,0}\rangle;&\langle T_{1,1}\rangle.\end{array}

\begin{array}[]{llllllll}\langle T_{1,0}+X_{\alpha}\rangle;&\langle T_{1,1}+X_{\beta}\rangle.\end{array}

\begin{array}[]{llllllll}\langle T_{1,0}+X_{\alpha}\rangle;&\langle T_{1,1}+X_{\beta}\rangle.\end{array}

\begin{array}[]{ccccccccccccc}C_{2}&\cong_{A_{1}}&\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle&\oplus&\langle X_{\beta},Y_{\alpha+\beta}\rangle&\oplus&\\ &&\langle X_{\alpha+\beta},Y_{\alpha+2\beta}\rangle&\oplus&\langle X_{\alpha}\rangle&\oplus&\\ &&\langle Y_{\alpha}\rangle&\oplus&\langle T_{0,1}\rangle\\ &\cong_{A_{1}}&V(2)&\oplus&V(1)&\oplus&\\ &&V(1)&\oplus&V(0)&\oplus&\\ &&V(0)&\oplus&V(0),\end{array}

\begin{array}[]{ccccccccccccc}C_{2}&\cong_{A_{1}}&\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle&\oplus&\langle X_{\beta},Y_{\alpha+\beta}\rangle&\oplus&\\ &&\langle X_{\alpha+\beta},Y_{\alpha+2\beta}\rangle&\oplus&\langle X_{\alpha}\rangle&\oplus&\\ &&\langle Y_{\alpha}\rangle&\oplus&\langle T_{0,1}\rangle\\ &\cong_{A_{1}}&V(2)&\oplus&V(1)&\oplus&\\ &&V(1)&\oplus&V(0)&\oplus&\\ &&V(0)&\oplus&V(0),\end{array}

\begin{array}[]{lllll}\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle T_{0,1}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha},T_{0,1}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha},T_{0,1}\rangle.\end{array}

\begin{array}[]{lllll}\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle T_{0,1}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha},T_{0,1}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha}\rangle,\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha},T_{0,1}\rangle.\end{array}

\begin{array}[]{lllllll}\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha}\rangle&\cong&A_{1}\oplus\mathfrak{n}_{1,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle T_{0,1}\rangle&\cong&A_{1}\oplus\mathfrak{n}_{1,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha},T_{0,1}\rangle&\cong&A_{1}\oplus\mathfrak{s}_{2,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha}\rangle&\cong&A_{1}\inplus\mathfrak{n}_{3,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha},T_{0,1}\rangle&\cong&A_{1}\inplus\mathfrak{s}_{4,8,A=1}.\end{array}

\begin{array}[]{lllllll}\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha}\rangle&\cong&A_{1}\oplus\mathfrak{n}_{1,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle T_{0,1}\rangle&\cong&A_{1}\oplus\mathfrak{n}_{1,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\oplus\langle X_{\alpha},T_{0,1}\rangle&\cong&A_{1}\oplus\mathfrak{s}_{2,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha}\rangle&\cong&A_{1}\inplus\mathfrak{n}_{3,1},\\ \langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle\inplus\langle X_{\beta},Y_{\alpha+\beta},Y_{\alpha},T_{0,1}\rangle&\cong&A_{1}\inplus\mathfrak{s}_{4,8,A=1}.\end{array}

\bBigg@{4}\{\begin{array}[]{llllllll}\displaystyle H_{\alpha},&H_{\beta},\\ \displaystyle X_{\alpha},&X_{\beta},&X_{\alpha+\beta},\\ \displaystyle Y_{\alpha},&Y_{\beta},&Y_{\alpha+\beta}\end{array}\bBigg@{4}\},

\bBigg@{4}\{\begin{array}[]{llllllll}\displaystyle H_{\alpha},&H_{\beta},\\ \displaystyle X_{\alpha},&X_{\beta},&X_{\alpha+\beta},\\ \displaystyle Y_{\alpha},&Y_{\beta},&Y_{\alpha+\beta}\end{array}\bBigg@{4}\},

\begin{array}[]{lllllllllllll}H_{\alpha}&=&\left(\begin{array}[]{rrr}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right),&H_{\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&1&0\\ 0&0&-1\end{array}\right),\\ \\ X_{\alpha}&=&\left(\begin{array}[]{rrr}0&1&0\\ 0&0&0\\ 0&0&0\end{array}\right),&Y_{\alpha}&=&\left(\begin{array}[]{rrr}0&0&0\\ 1&0&0\\ 0&0&0\end{array}\right),\\ \\ X_{\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&0&1\\ 0&0&0\end{array}\right),&Y_{\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&0&0\\ 0&1&0\end{array}\right),\\ \\ X_{\alpha+\beta}&=&\left(\begin{array}[]{rrr}0&0&-1\\ 0&0&0\\ 0&0&0\end{array}\right),&Y_{\alpha+\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&0&0\\ -1&0&0\end{array}\right).\end{array}

\begin{array}[]{lllllllllllll}H_{\alpha}&=&\left(\begin{array}[]{rrr}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right),&H_{\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&1&0\\ 0&0&-1\end{array}\right),\\ \\ X_{\alpha}&=&\left(\begin{array}[]{rrr}0&1&0\\ 0&0&0\\ 0&0&0\end{array}\right),&Y_{\alpha}&=&\left(\begin{array}[]{rrr}0&0&0\\ 1&0&0\\ 0&0&0\end{array}\right),\\ \\ X_{\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&0&1\\ 0&0&0\end{array}\right),&Y_{\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&0&0\\ 0&1&0\end{array}\right),\\ \\ X_{\alpha+\beta}&=&\left(\begin{array}[]{rrr}0&0&-1\\ 0&0&0\\ 0&0&0\end{array}\right),&Y_{\alpha+\beta}&=&\left(\begin{array}[]{rrr}0&0&0\\ 0&0&0\\ -1&0&0\end{array}\right).\end{array}

\begin{array}[]{llllllllllllll}\{H_{\alpha},~{}H_{\beta},~{}X_{\alpha},~{}X_{\beta},~{}Y_{\alpha},~{}Y_{\beta}\},\end{array}

\begin{array}[]{llllllllllllll}\{H_{\alpha},~{}H_{\beta},~{}X_{\alpha},~{}X_{\beta},~{}Y_{\alpha},~{}Y_{\beta}\},\end{array}

\begin{array}[]{llllllllll}H_{\alpha}&=&\left(\begin{array}[]{rrrr}1&0&0&0\\ 0&-1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right),&H_{\beta}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&-1\end{array}\right),\\ \\ X_{\alpha}&=&\left(\begin{array}[]{rrrr}0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right),&Y_{\alpha}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right),\\ \\ X_{\beta}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{array}\right),&Y_{\beta}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\end{array}\right).\end{array}

\begin{array}[]{llllllllll}H_{\alpha}&=&\left(\begin{array}[]{rrrr}1&0&0&0\\ 0&-1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right),&H_{\beta}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&-1\end{array}\right),\\ \\ X_{\alpha}&=&\left(\begin{array}[]{rrrr}0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right),&Y_{\alpha}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right),\\ \\ X_{\beta}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{array}\right),&Y_{\beta}&=&\left(\begin{array}[]{rrrr}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&1&0\end{array}\right).\end{array}

Φ^{+} = {α, β, α + β, 2 α + β, 3 α + β, 3 α + 2 β} .

Φ^{+} = {α, β, α + β, 2 α + β, 3 α + β, 3 α + 2 β} .

\bBigg@{4}\{\begin{array}[]{llllllll}\displaystyle H_{\alpha},&H_{\beta},\\ \displaystyle X_{\alpha},&X_{\beta},&X_{\alpha+\beta},&X_{2\alpha+\beta},&X_{3\alpha+\beta},&X_{3\alpha+2\beta},\\ \displaystyle Y_{\alpha},&Y_{\beta},&Y_{\alpha+\beta},&Y_{2\alpha+\beta},&Y_{3\alpha+\beta},&Y_{3\alpha+2\beta}\end{array}\bBigg@{4}\},

\bBigg@{4}\{\begin{array}[]{llllllll}\displaystyle H_{\alpha},&H_{\beta},\\ \displaystyle X_{\alpha},&X_{\beta},&X_{\alpha+\beta},&X_{2\alpha+\beta},&X_{3\alpha+\beta},&X_{3\alpha+2\beta},\\ \displaystyle Y_{\alpha},&Y_{\beta},&Y_{\alpha+\beta},&Y_{2\alpha+\beta},&Y_{3\alpha+\beta},&Y_{3\alpha+2\beta}\end{array}\bBigg@{4}\},

\begin{array}[]{lllllllllll}\displaystyle[H_{\mu},X_{\nu}]=\frac{2(\mu,\nu)}{(\mu,\mu)}X_{\nu},\quad[X_{\mu},X_{-\mu}]=H_{\mu},~{}\mu,\nu\in\Phi,\\ \displaystyle[X_{\mu},X_{\nu}]=N_{\mu,\nu}X_{\mu+\nu},\quad\mu,\nu,\mu+\nu\in\Phi,\\ \displaystyle 1=N_{\beta,\alpha}=N_{\beta,3\alpha+\beta}=N_{3\alpha+\beta,-(3\alpha+2\beta)}=N_{2\alpha+\beta,-(3\alpha+\beta)}=\\ N_{2\alpha+\beta,-(3\alpha+2\beta)}=N_{-(3\alpha+2\beta),\alpha+\beta}=N_{-(3\alpha+2\beta),\beta}=N_{-(3\alpha+\beta),\alpha}=\\ \displaystyle N_{-(\alpha+\beta),\beta},\\ 2=N_{\alpha+\beta,\alpha}=N_{\alpha,-(2\alpha+\beta)}=N_{-(2\alpha+\beta),\alpha+\beta},\\ 3=N_{\alpha,2\alpha+\beta}=N_{\alpha,-(\alpha+\beta)}=N_{\alpha+\beta,2\alpha+\beta},\\ N_{\mu,\nu}=-N_{\nu,\mu}=-N_{-\mu,-\nu},\end{array}

\begin{array}[]{lllllllllll}\displaystyle[H_{\mu},X_{\nu}]=\frac{2(\mu,\nu)}{(\mu,\mu)}X_{\nu},\quad[X_{\mu},X_{-\mu}]=H_{\mu},~{}\mu,\nu\in\Phi,\\ \displaystyle[X_{\mu},X_{\nu}]=N_{\mu,\nu}X_{\mu+\nu},\quad\mu,\nu,\mu+\nu\in\Phi,\\ \displaystyle 1=N_{\beta,\alpha}=N_{\beta,3\alpha+\beta}=N_{3\alpha+\beta,-(3\alpha+2\beta)}=N_{2\alpha+\beta,-(3\alpha+\beta)}=\\ N_{2\alpha+\beta,-(3\alpha+2\beta)}=N_{-(3\alpha+2\beta),\alpha+\beta}=N_{-(3\alpha+2\beta),\beta}=N_{-(3\alpha+\beta),\alpha}=\\ \displaystyle N_{-(\alpha+\beta),\beta},\\ 2=N_{\alpha+\beta,\alpha}=N_{\alpha,-(2\alpha+\beta)}=N_{-(2\alpha+\beta),\alpha+\beta},\\ 3=N_{\alpha,2\alpha+\beta}=N_{\alpha,-(\alpha+\beta)}=N_{\alpha+\beta,2\alpha+\beta},\\ N_{\mu,\nu}=-N_{\nu,\mu}=-N_{-\mu,-\nu},\end{array}

\begin{array}[]{llll}\mathfrak{n}_{1,1}&\text{The abelian Lie algebra of dimension $1$}\\ \end{array}

\begin{array}[]{llll}\mathfrak{n}_{1,1}&\text{The abelian Lie algebra of dimension $1$}\\ \end{array}

\begin{array}[]{llll}\mathfrak{s}_{2,1}&[e_{2},e_{1}]=e_{1}\end{array}

\begin{array}[]{llll}\mathfrak{s}_{2,1}&[e_{2},e_{1}]=e_{1}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{3,1}&[e_{2},e_{3}]=e_{1}\\ \mathfrak{s}_{3,1}&[e_{3},e_{1}]=e_{1},[e_{3},e_{2}]=Ae_{2}\\ &0<|A|\leq 1,~{}\text{if}~{}|A|=1~{}\text{then}~{}\arg(A)\leq\pi\\ \mathfrak{s}_{3,2}&[e_{3},e_{1}]=e_{1},[e_{3},e_{2}]=e_{1}+e_{2}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{3,1}&[e_{2},e_{3}]=e_{1}\\ \mathfrak{s}_{3,1}&[e_{3},e_{1}]=e_{1},[e_{3},e_{2}]=Ae_{2}\\ &0<|A|\leq 1,~{}\text{if}~{}|A|=1~{}\text{then}~{}\arg(A)\leq\pi\\ \mathfrak{s}_{3,2}&[e_{3},e_{1}]=e_{1},[e_{3},e_{2}]=e_{1}+e_{2}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{4,1}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2}\\ \mathfrak{s}_{4,2}&[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{1}+e_{2},[e_{4},e_{3}]=e_{2}+e_{3}\\ \mathfrak{s}_{4,3}&[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=Ae_{2},[e_{4},e_{3}]=Be_{3},\\ &0<|B|\leq|A|\leq 1,(A,B)\neq(-1,-1)\\ \mathfrak{s}_{4,4}&[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{1}+e_{2},[e_{4},e_{3}]=Ae_{3},\\ &A\neq 0\\ \mathfrak{s}_{4,6}&[e_{2},e_{3}]=e_{1},[e_{4},e_{2}]=e_{2},[e_{4},e_{3}]=-e_{3}\\ \mathfrak{s}_{4,8}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=(1+A)e_{1},[e_{4},e_{2}]=e_{2},[e_{4},e_{3}]=Ae_{3},\\ &0<|A|\leq 1,~{}\text{if}~{}|A|=1~{}\text{then}~{}\arg(A)<\pi\\ \mathfrak{s}_{4,10}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=2e_{1},[e_{4},e_{2}]=e_{2},[e_{4},e_{3}]=e_{2}+e_{3}\\ \mathfrak{s}_{4,11}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{2}\\ \mathfrak{s}_{4,12}&[e_{3},e_{1}]=e_{1},[e_{3},e_{2}]=e_{2},[e_{4},e_{1}]=-e_{2},[e_{4},e_{2}]=e_{1}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{4,1}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2}\\ \mathfrak{s}_{4,2}&[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{1}+e_{2},[e_{4},e_{3}]=e_{2}+e_{3}\\ \mathfrak{s}_{4,3}&[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=Ae_{2},[e_{4},e_{3}]=Be_{3},\\ &0<|B|\leq|A|\leq 1,(A,B)\neq(-1,-1)\\ \mathfrak{s}_{4,4}&[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{1}+e_{2},[e_{4},e_{3}]=Ae_{3},\\ &A\neq 0\\ \mathfrak{s}_{4,6}&[e_{2},e_{3}]=e_{1},[e_{4},e_{2}]=e_{2},[e_{4},e_{3}]=-e_{3}\\ \mathfrak{s}_{4,8}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=(1+A)e_{1},[e_{4},e_{2}]=e_{2},[e_{4},e_{3}]=Ae_{3},\\ &0<|A|\leq 1,~{}\text{if}~{}|A|=1~{}\text{then}~{}\arg(A)<\pi\\ \mathfrak{s}_{4,10}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=2e_{1},[e_{4},e_{2}]=e_{2},[e_{4},e_{3}]=e_{2}+e_{3}\\ \mathfrak{s}_{4,11}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{2}\\ \mathfrak{s}_{4,12}&[e_{3},e_{1}]=e_{1},[e_{3},e_{2}]=e_{2},[e_{4},e_{1}]=-e_{2},[e_{4},e_{2}]=e_{1}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{5,2}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3}\\ \mathfrak{n}_{5,3}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1}\\ \mathfrak{n}_{5,6}&[e_{2},e_{5}]=e_{1},[e_{3},e_{4}]=e_{1},[e_{3},e_{5}]=e_{2},[e_{4},e_{5}]=e_{3}\\ \mathfrak{s}_{5,17}&[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=-e_{3},[e_{5},e_{4}]=Ae_{4},\\ &[e_{2},e_{3}]=e_{1},\\ &0\leq\mathrm{Re}(A),~{}\text{if}~{}\mathrm{Re}(A)=0~{}\text{then}~{}0<\mathrm{Im}(A)\\ \mathfrak{s}_{5,21}&[e_{5},e_{1}]=2e_{1},[e_{5},e_{2}]=e_{2}+e_{3},[e_{5},e_{3}]=e_{3}+e_{4}\\ &[e_{5},e_{4}]=e_{4},[e_{2},e_{3}]=e_{1}\\ \mathfrak{s}_{5,22}&[e_{5},e_{1}]=(A+1)e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=Ae_{3},[e_{5},e_{4}]=Be_{4},\\ &[e_{2},e_{3}]=e_{1},\\ &0<|A|\leq 1,B\neq 0,~{}\text{if}~{}|A|=1~{}\text{then}~{}\mathrm{arg}(A)<\pi\\ \mathfrak{s}_{5,26}&[e_{5},e_{1}]=(A+1)e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=Ae_{3},\\ &[e_{5},e_{4}]=e_{1}+(A+1)e_{4},[e_{2},e_{3}]=e_{1},\\ &0<|A|\leq 1,~{}\text{if}~{}|A|=1~{}\text{then}~{}\mathrm{arg}(A)<\pi\\ \mathfrak{s}_{5,30}&[e_{5},e_{1}]=e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=Ae_{4},\\ &[e_{2},e_{3}]=e_{1},\\ &A\neq 0\\ \mathfrak{s}_{5,33}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{2}]=-e_{2},\\ &[e_{5},e_{3}]=-2e_{3},[e_{5},e_{4}]=e_{4}\\ \mathfrak{s}_{5,35}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=(A+2)e_{1},\\ &[e_{5},e_{2}]=(A+1)e_{2},[e_{5},e_{3}]=Ae_{3},[e_{5},e_{4}]=e_{4},\\ &A\neq 0,-2\\ \mathfrak{s}_{5,36}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=2e_{1},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=e_{4}\\ \mathfrak{s}_{5,37}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=e_{1},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=e_{3}\\ \mathfrak{s}_{5,41}&[e_{4},e_{1}]=e_{1},[e_{4},e_{3}]=Ae_{3},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=Be_{3},\\ &0<|B|\leq|A|\leq 1\\ \mathfrak{s}_{5,44}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{2},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=-e_{3}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{5,2}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3}\\ \mathfrak{n}_{5,3}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1}\\ \mathfrak{n}_{5,6}&[e_{2},e_{5}]=e_{1},[e_{3},e_{4}]=e_{1},[e_{3},e_{5}]=e_{2},[e_{4},e_{5}]=e_{3}\\ \mathfrak{s}_{5,17}&[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=-e_{3},[e_{5},e_{4}]=Ae_{4},\\ &[e_{2},e_{3}]=e_{1},\\ &0\leq\mathrm{Re}(A),~{}\text{if}~{}\mathrm{Re}(A)=0~{}\text{then}~{}0<\mathrm{Im}(A)\\ \mathfrak{s}_{5,21}&[e_{5},e_{1}]=2e_{1},[e_{5},e_{2}]=e_{2}+e_{3},[e_{5},e_{3}]=e_{3}+e_{4}\\ &[e_{5},e_{4}]=e_{4},[e_{2},e_{3}]=e_{1}\\ \mathfrak{s}_{5,22}&[e_{5},e_{1}]=(A+1)e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=Ae_{3},[e_{5},e_{4}]=Be_{4},\\ &[e_{2},e_{3}]=e_{1},\\ &0<|A|\leq 1,B\neq 0,~{}\text{if}~{}|A|=1~{}\text{then}~{}\mathrm{arg}(A)<\pi\\ \mathfrak{s}_{5,26}&[e_{5},e_{1}]=(A+1)e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=Ae_{3},\\ &[e_{5},e_{4}]=e_{1}+(A+1)e_{4},[e_{2},e_{3}]=e_{1},\\ &0<|A|\leq 1,~{}\text{if}~{}|A|=1~{}\text{then}~{}\mathrm{arg}(A)<\pi\\ \mathfrak{s}_{5,30}&[e_{5},e_{1}]=e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=Ae_{4},\\ &[e_{2},e_{3}]=e_{1},\\ &A\neq 0\\ \mathfrak{s}_{5,33}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{2}]=-e_{2},\\ &[e_{5},e_{3}]=-2e_{3},[e_{5},e_{4}]=e_{4}\\ \mathfrak{s}_{5,35}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=(A+2)e_{1},\\ &[e_{5},e_{2}]=(A+1)e_{2},[e_{5},e_{3}]=Ae_{3},[e_{5},e_{4}]=e_{4},\\ &A\neq 0,-2\\ \mathfrak{s}_{5,36}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=2e_{1},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=e_{4}\\ \mathfrak{s}_{5,37}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=e_{1},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=e_{3}\\ \mathfrak{s}_{5,41}&[e_{4},e_{1}]=e_{1},[e_{4},e_{3}]=Ae_{3},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=Be_{3},\\ &0<|B|\leq|A|\leq 1\\ \mathfrak{s}_{5,44}&[e_{2},e_{3}]=e_{1},[e_{4},e_{1}]=e_{1},[e_{4},e_{2}]=e_{2},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{3}]=-e_{3}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{6,18}&[e_{2},e_{6}]=e_{1},[e_{3},e_{4}]=e_{1},[e_{3},e_{5}]=e_{2},[e_{4},e_{5}]=e_{3},[e_{5},e_{6}]=e_{4}\\ \mathfrak{s}_{6,150}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{2}]=3e_{2},\\ &[e_{6},e_{3}]=e_{3},[e_{6},e_{4}]=2e_{4},[e_{6},e_{5}]=-e_{5}\\ \mathfrak{s}_{6,153}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{1}]=3e_{1},\\ &[e_{6},e_{2}]=e_{1}+3e_{2},[e_{6},e_{3}]=2e_{3},[e_{6},e_{4}]=e_{4}+e_{5},[e_{6},e_{5}]=e_{5}\\ \mathfrak{s}_{6,155}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{1}]=(2A+1)e_{1},\\ &[e_{6},e_{2}]=(A+2)e_{2},[e_{6},e_{3}]=(A+1)e_{3},[e_{6},e_{4}]=e_{4},\\ &[e_{6},e_{5}]=Ae_{5},\\ &0<|A|\leq 1,A\neq-1/2,\mathrm{arg}(A)\leq\pi~{}\text{if}~{}|A|=1\\ \mathfrak{s}_{6,156}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{1}]=e_{1},\\ &[e_{6},e_{2}]=2e_{2},[e_{6},e_{3}]=e_{3},[e_{6},e_{4}]=e_{4}\\ \mathfrak{s}_{6,162}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{2}]=e_{2},[e_{6},e_{3}]=Ae_{3}\\ &[e_{6},e_{4}]=-e_{4},[e_{6},e_{5}]=-Ae_{5},\\ &0<|A|\leq 1~{}\text{and}~{}\mathrm{arg}(A)<\pi,~{}\text{if}~{}|A|=1~{}\text{then}~{}\mathrm{arg}(A)<\pi/2\\ \mathfrak{s}_{6,168}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{1}]=2e_{1},[e_{6},e_{2}]=(A+1)e_{2},\\ &[e_{6},e_{3}]=(B+1)e_{3},[e_{6},e_{4}]=(1-A)e_{4},[e_{6},e_{5}]=(1-B)e_{5},\\ &|B|\leq|A|,~{}\mathrm{arg}(A),\mathrm{arg}(B)<\pi,A,B\neq 1.\\ &\text{If}~{}|A|=|B|~{}\text{then}~{}\mathrm{arg}(B)\leq\mathrm{arg}(A).\\ \mathfrak{s}_{6,171}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{1}]=2e_{1},[e_{6},e_{2}]=e_{2}+e_{3},\\ &[e_{6},e_{3}]=e_{3}-e_{5},[e_{6},e_{4}]=e_{4},[e_{6},e_{5}]=-e_{4}+e_{5}\\ \mathfrak{s}_{6,178}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{1}]=2e_{1},[e_{6},e_{2}]=(1+A)e_{2},\\ &[e_{6},e_{3}]=2e_{3},[e_{6},e_{4}]=(1-A)e_{4},\\ &A\neq 1,\mathrm{arg}(A)<\pi\\ \mathfrak{s}_{6,197}&[e_{2},e_{5}]=e_{1},[e_{3},e_{4}]=e_{1},[e_{3},e_{5}]=e_{2},[e_{4},e_{5}]=e_{3},\\ &[e_{6},e_{1}]=5e_{1},[e_{6},e_{2}]=4e_{2},[e_{6},e_{3}]=3e_{3},[e_{6},e_{4}]=2e_{4},\\ &[e_{6},e_{5}]=e_{5}\\ \mathfrak{s}_{6,234}&[e_{2},e_{3}]=e_{1},[e_{5},e_{1}]=e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=Ae_{4}\\ &[e_{6},e_{1}]=e_{1},[e_{6},e_{3}]=e_{3},[e_{6},e_{4}]=Be_{4},\\ &|A|\leq|B|,B\neq 0.~{}\text{If}~{}|A|=|B|~{}\text{then}~{}\mathrm{arg}(A)\leq\mathrm{arg}(B)\\ \mathfrak{s}_{6,242}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=2e_{1},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=e_{4},[e_{6},e_{1}]=e_{1},\\ &[e_{6},e_{2}]=e_{2},[e_{6},e_{3}]=e_{3}\end{array}

\begin{array}[]{llll}\mathfrak{n}_{6,18}&[e_{2},e_{6}]=e_{1},[e_{3},e_{4}]=e_{1},[e_{3},e_{5}]=e_{2},[e_{4},e_{5}]=e_{3},[e_{5},e_{6}]=e_{4}\\ \mathfrak{s}_{6,150}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{2}]=3e_{2},\\ &[e_{6},e_{3}]=e_{3},[e_{6},e_{4}]=2e_{4},[e_{6},e_{5}]=-e_{5}\\ \mathfrak{s}_{6,153}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{1}]=3e_{1},\\ &[e_{6},e_{2}]=e_{1}+3e_{2},[e_{6},e_{3}]=2e_{3},[e_{6},e_{4}]=e_{4}+e_{5},[e_{6},e_{5}]=e_{5}\\ \mathfrak{s}_{6,155}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{1}]=(2A+1)e_{1},\\ &[e_{6},e_{2}]=(A+2)e_{2},[e_{6},e_{3}]=(A+1)e_{3},[e_{6},e_{4}]=e_{4},\\ &[e_{6},e_{5}]=Ae_{5},\\ &0<|A|\leq 1,A\neq-1/2,\mathrm{arg}(A)\leq\pi~{}\text{if}~{}|A|=1\\ \mathfrak{s}_{6,156}&[e_{3},e_{4}]=e_{2},[e_{3},e_{5}]=e_{1},[e_{4},e_{5}]=e_{3},[e_{6},e_{1}]=e_{1},\\ &[e_{6},e_{2}]=2e_{2},[e_{6},e_{3}]=e_{3},[e_{6},e_{4}]=e_{4}\\ \mathfrak{s}_{6,162}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{2}]=e_{2},[e_{6},e_{3}]=Ae_{3}\\ &[e_{6},e_{4}]=-e_{4},[e_{6},e_{5}]=-Ae_{5},\\ &0<|A|\leq 1~{}\text{and}~{}\mathrm{arg}(A)<\pi,~{}\text{if}~{}|A|=1~{}\text{then}~{}\mathrm{arg}(A)<\pi/2\\ \mathfrak{s}_{6,168}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{1}]=2e_{1},[e_{6},e_{2}]=(A+1)e_{2},\\ &[e_{6},e_{3}]=(B+1)e_{3},[e_{6},e_{4}]=(1-A)e_{4},[e_{6},e_{5}]=(1-B)e_{5},\\ &|B|\leq|A|,~{}\mathrm{arg}(A),\mathrm{arg}(B)<\pi,A,B\neq 1.\\ &\text{If}~{}|A|=|B|~{}\text{then}~{}\mathrm{arg}(B)\leq\mathrm{arg}(A).\\ \mathfrak{s}_{6,171}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{1}]=2e_{1},[e_{6},e_{2}]=e_{2}+e_{3},\\ &[e_{6},e_{3}]=e_{3}-e_{5},[e_{6},e_{4}]=e_{4},[e_{6},e_{5}]=-e_{4}+e_{5}\\ \mathfrak{s}_{6,178}&[e_{2},e_{4}]=e_{1},[e_{3},e_{5}]=e_{1},[e_{6},e_{1}]=2e_{1},[e_{6},e_{2}]=(1+A)e_{2},\\ &[e_{6},e_{3}]=2e_{3},[e_{6},e_{4}]=(1-A)e_{4},\\ &A\neq 1,\mathrm{arg}(A)<\pi\\ \mathfrak{s}_{6,197}&[e_{2},e_{5}]=e_{1},[e_{3},e_{4}]=e_{1},[e_{3},e_{5}]=e_{2},[e_{4},e_{5}]=e_{3},\\ &[e_{6},e_{1}]=5e_{1},[e_{6},e_{2}]=4e_{2},[e_{6},e_{3}]=3e_{3},[e_{6},e_{4}]=2e_{4},\\ &[e_{6},e_{5}]=e_{5}\\ \mathfrak{s}_{6,234}&[e_{2},e_{3}]=e_{1},[e_{5},e_{1}]=e_{1},[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=Ae_{4}\\ &[e_{6},e_{1}]=e_{1},[e_{6},e_{3}]=e_{3},[e_{6},e_{4}]=Be_{4},\\ &|A|\leq|B|,B\neq 0.~{}\text{If}~{}|A|=|B|~{}\text{then}~{}\mathrm{arg}(A)\leq\mathrm{arg}(B)\\ \mathfrak{s}_{6,242}&[e_{2},e_{4}]=e_{1},[e_{3},e_{4}]=e_{2},[e_{5},e_{1}]=2e_{1},\\ &[e_{5},e_{2}]=e_{2},[e_{5},e_{4}]=e_{4},[e_{6},e_{1}]=e_{1},\\ &[e_{6},e_{2}]=e_{2},[e_{6},e_{3}]=e_{3}\end{array}

g = p \inplus r, where [p, p] = p, [r, r] \subset r, [p, r] \subseteq r .

g = p \inplus r, where [p, p] = p, [r, r] \subset r, [p, r] \subseteq r .

[h, p_{\pm}] = \pm 2 p_{\pm}, [p_{+}, p_{-}] = h .

[h, p_{\pm}] = \pm 2 p_{\pm}, [p_{+}, p_{-}] = h .

\begin{array}[]{lllll}\rho_{j}(h)v_{k}=(j-1-2k)v_{k},~{}\rho_{j}(p_{-})v_{k}=v_{k+1},\\ \rho_{j}(p_{+})v_{k}=k(j-1)v_{k-1},k\in\{0,1,...,j-1\}.\end{array}

\begin{array}[]{lllll}\rho_{j}(h)v_{k}=(j-1-2k)v_{k},~{}\rho_{j}(p_{-})v_{k}=v_{k+1},\\ \rho_{j}(p_{+})v_{k}=k(j-1)v_{k-1},k\in\{0,1,...,j-1\}.\end{array}

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Full text

Subalgebras of the rank two semisimple Lie algebras

Andrew Douglas1,2,3

1Ph.D. Programs in Mathematics and Physics, The Graduate Center, City University of New York, New York, NY, 10016, USA

2Department of Mathematics, New York City College of Technology, City University of New York, Brooklyn, NY, 11201, USA

and

Joe Repka3

3Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

Abstract.

In this expository article, we describe the classification of the subalgebras of the rank $2$ semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the subalgebras of the classical rank 2 semisimple Lie algebras. Finally, Mayanskiy finished the classification of the subalgebras of the remaining rank $2$ semisimple Lie algebra, the exceptional Lie algebra $G_{2}$ . We identify subalgebras of the classification in terms of a uniform classification scheme of Lie algebras of low dimension. The classification is up to inner automorphism, and the ground field is the complex numbers.

Key words and phrases:

Semisimple Lie algebras, solvable Lie algebras, Levi decomposable algebras, Lie subalgebras

2010 Mathematics Subject Classification:

17B05, 17B10, 17B20, 17B25, 17B30

1 Introduction
2 The subalgebras of $C_{2}$
2.1 Classifying and identifying the solvable subalgebras of $C_{2}$
2.2 Classifying and identifying the Levi decomposable subalgebras of $C_{2}$
3 The subalgebras of $A_{2}$
4 The subalgebras of $A_{1}\times A_{1}$
5 The subalgebras of $G_{2}$
A Classification of Lie algebras
A.1 Classification of semisimple Lie algebras
A.2 Classification of solvable Lie algebras
A.3 Classification of Levi decomposable algebras

List of Tables

1 One-dimensional (solvable) subalgebras of $C_{2}$
2 Two-dimensional (solvable) subalgebras of $C_{2}$
3 Three-dimensional solvable subalgebras of $C_{2}$
4 Four-dimensional solvable subalgebras of $C_{2}$
5 Five- and six-dimensional solvable subalgebras of $C_{2}$
6 Semisimple and Levi decomposable subalgebras of $C_{2}$
7 One- and two-dimensional (solvable) subalgebras of $A_{2}$
8 Three-, four-, and five-dimensional solvable subalgebras of $A_{2}$
9 Semisimple and Levi decomposable subalgebras of $A_{2}$
10 Solvable subalgebras of $A_{1}\times A_{1}$
11 Semisimple and Levi decomposable subalgebras of $A_{1}\times A_{1}$
12 One-dimensional (solvable) subalgebras of $G_{2}$
13 Two-dimensional (solvable) subalgebras of $G_{2}$
14 Regular three-dimensional solvable subalgebras of $G_{2}$
15 Non-regular three-dimensional solvable subalgebras of $G_{2}$
16 Regular four-dimensional solvable subalgebras of $G_{2}$
17 Non-regular four-dimensional solvable subalgebras of $G_{2}$
18 Five-dimensional solvable subalgebras of $G_{2}$
19 Six-dimensional solvable subalgebras of $G_{2}$
20 Seven- and eight-dimensional solvable subalgebras of $G_{2}$
21 Semisimple subalgebras of $G_{2}$
22 Levi decomposable subalgebras of $G_{2}$

1. Introduction

In this expository article, we describe the classification of the subalgebras of the rank $2$ semisimple Lie algebras. Specifically, we describe the subalgebras of the symplectic algebra $C_{2}$ , the special linear algebra $A_{2}$ , the semisimple–and not simple–Lie algebra $A_{1}\times A_{1}$ , and the exceptional Lie algebra $G_{2}$ . The classification is up to inner automorphism, and the ground field is the complex numbers.

We omit virtually all proofs in this article, as they may be found in the literature, which we cite. But, we do outline techniques, strategy, and theory employed in proofs for one case–the subalgebras of $C_{2}$ .

By Levi’s Theorem [[ŠW14], Chapter II, Section 2], a subalgebra of a complex semisimple Lie algebra is either semisimple, solvable, or a nontrivial semidirect sum of the first two. A subalgebra that is a nontrivial semidirect sum of a semisimple subalgebra with a solvable subalgebra is called a Levi decomposable subalgebra.

Semisimple subalgebras of semisimple Lie algebras have been extensively studied [dGr11, Dyn52a, Dyn52b, LG72, Min06]. For instance, the semisimple subalgebras of the exceptional Lie algebras have been classified, up to inner automorphism [Min06]. As another important example, de Graaf [dGr11] classified the semisimple subalgebras of the simple Lie algebras of ranks $\leq 8$ . In particular, the semisimple subalgebras of the rank $2$ simple Lie algebras have been known for some time.

Until quite recently, much less research had examined and classified the non-semisimple subalgebras of semisimple Lie algebras. We recently classified the solvable and Levi decomposable subalgebras of the symplectic algebra $C_{2}$ [DR15, DR17] and the special Linear algebra $A_{2}$ [DR16b]. We also gave a full classification of the subalgebras of the semisimple Lie algebra $A_{1}\times A_{1}$ [DR16a]. In $2016$ , Mayanskiy posted a classification of the subalgebras of $G_{2}$ [May16]. The aforementioned work completes the classification of the subalgebras of the rank $2$ semisimple Lie algebras and a description of this classification, and identification thereof, encompasses the present article.

Interestingly, in Mayanskiy’s [May16] classification of $G_{2}$ subalgebras, there are naturally parameterized families of inequivalent subalgebras for which subalgebras of the family are not only inequivalent (under inner automorphisms), but are also non-isomorphic. This phenomenon also occurs in the classification of $A_{2}$ and $C_{2}$ subalgebras [DR16b, DR17], where we precisely describe the different isomorphism classes within such a family. In this article, we also identify such families for $G_{2}$ , and the non-isomorphic classes therein.

When possible, we identify solvable and Levi decomposable subalgebras in terms of the (partial) classification of solvable and Levi decomposable algebras presented by Šnobl and Winternitz [ŠW14]. Further, we describe the pertinent portion of Šnobl and Winternitz’s classification in the appendices of the present article.

Our rationale for writing this article is threefold. First, we felt it beneficial to students and researchers in the field to have the entire subalgebra classification in one well-organized article. Second, we wanted to describe the classification using one consistent choice of notation, which is not done at present in the literature on subalgebras. Thirdly, we wanted to identify the subalgebras with respect to one classification, which was not previously done in totality. Specifically, identification was not available for the subalgebras of $G_{2}$ .

We end the section by briefly articulating the importance of the mathematical domain of this article–subalgebras of Lie algebras. In addition to the intrinsic mathematical significance of classifications of subalgebras of Lie algebras (or classifications of subgroups of corresponding Lie groups), such classifications also have physical significance and mathematical applications, some of which are listed below (as we described in [DR17]):

•

If a system of differential equations is invariant under a Lie group, then its subgroups can be used to construct group invariant solutions [Olv86].

•

Subgroups of the symmetry groups of nonlinear partial differential equations provide a method for performing symmetry reduction (reducing the number of independent variables) [ORWZ90, GHP84, DKLW86].

•

A knowledge of the subgroup structure of a Lie group $G$ is needed if we are interested in considering all possible contractions of $G$ to other groups [PW77].

•

Physical models–such as the vibron model, and the interacting boson model– use chains of subalgebras, and these subalgebras need to be explicitly described in applications [IA87].

2. The subalgebras of $C_{2}$

The simple Lie algebra $C_{2}$ may be realized as the symplectic algebra $\mathfrak{sp}(4,\mathbb{C})$ of $4\times 4$ complex matrices $X$ satisfying $JX^{t}J=X$ , where $J$ is the matrix

[TABLE]

The root system associated with $C_{2}$ is depicted in Figure 1.

It has positive roots $\alpha,~{}\beta,~{}\alpha+\beta$ , and $\alpha+2\beta$ . Under the identification with $\mathfrak{sp}(4,\mathbb{C})$ , $C_{2}$ has a Chevalley basis

[TABLE]

where

[TABLE]

and

[TABLE]

$H_{\alpha}$ and $H_{\beta}$ are the coroots; $X_{\alpha},X_{\beta},X_{\alpha+\beta}$ , and $X_{\alpha+2\beta}$ are the positive root vectors; and $Y_{\alpha},Y_{\beta},Y_{\alpha+\beta}$ , and $Y_{\alpha+2\beta}$ are the negative root vectors. The classification becomes somewhat less cumbersome if we also make the definition:

[TABLE]

We highlight three distinguished subalgebras: A Cartan subalgebra $\mathfrak{h}=\langle H_{\alpha},H_{\beta}\rangle$ ; a Borel subalgebra $\mathfrak{b}=\langle H_{\alpha},H_{\beta},X_{\alpha},X_{\beta},X_{\alpha+\beta},X_{\alpha+2\beta}\rangle$ ; and the nilradical of the Borel subalgebra $\mathfrak{n}=\langle X_{\alpha},X_{\beta},X_{\alpha+\beta},X_{\alpha+2\beta}\rangle$ .

The Lie group corresponding to $C_{2}$ , under its identification with $\mathfrak{sp}(4,\mathbb{C})$ , is the symplectic group $\mathrm{Sp}(4,\mathbb{C})$ . It is the Lie group of $4\times 4$ matrices $M$ with complex entries such that $MJM^{t}=J$ .

The classification and identification of the subalgebras of $C_{2}$ are described in tables below. The classification is from [DR15, DR17]. The classification is organized by dimension, and further organized by structure. Tables 1-5 contain the classification and identification of the solvable subalgebras, and Table 6 contains the classification and identification of the semisimple and Levi decomposable subalgebras.

In the following subsections, we will describe the techniques or strategies used in the classification of solvable subalgebras of $C_{2}$ and the Levi decomposable subalgebras of $C_{2}$ . We will only do this for the $C_{2}$ case, and we will omit virtually all proofs. We’ll also discuss identification of subalgebras with respect to the classification of solvable and Levi decomposable Lie algebras in [ŠW14].

2.1. Classifying and identifying the solvable subalgebras of $C_{2}$

We proceed by dimension to classify the solvable subalgebras of $C_{2}$ . For each dimension, we further subdivide the classification according to structure.

In dimension $1$ , we consider three subcases depending on whether the generator is semisimple, nilpotent, or has a non-trivial Jordan decomposition. If the generator is nilpotent, we may use the rich theory on nilpotent orbits that may be found in Collingwood and McGovern’s classic text [CM93]. Specifically, there are precisely three nonzero nilpotent orbits of $C_{2}$ [[CM93], Theorem 5.1.3] with representatives $X_{\alpha},X_{\beta}$ , and $X_{\alpha}+X_{\beta}$ , giving us the three possible inequivalent subalgebras with nilpotent generator

[TABLE]

In the case where the generator is semisimple, we first use the fact that every semisimple element $T$ of $C_{2}$ is conjugate to an element in $\mathfrak{h}$ [[CM93], Corollary 2.2.2], so we may assume $T\in\mathfrak{h}$ . We also use the result that two elements in $\mathfrak{t}$ are $\mathrm{Sp}(4,\mathbb{C})$ -conjugate if and only if they are $W$ -conjugate [[CM93], Theorem 2.2.4], where $W$ is the Weyl group corresponding to $\mathfrak{h}$ . The Weyl group $W$ of $C_{2}$ has generators $s_{\alpha}$ and $s_{\beta}$ such that $s_{\alpha}(T_{a,b})=T_{a,-b}$ and $s_{\beta}(T_{a,b})=T_{b,a}$ . It follows that a complete list of one-dimensional subalgebras of $C_{2}$ with semisimple generators is

[TABLE]

We then rely primarily on linear algebra to consider the case where the generator has a non-trivial Jordan decomposition. That is, the generator is the sum of a non-zero nilpotent element and a non-zero semisimple element that commutes with it. This yields a complete list of one-dimensional subalgebras of $C_{2}$ with generators having a nontrivial Jordan decomposition:

[TABLE]

Eqs. (7)-(9) give us a complete classification of one-dimensional subalgebras of $C_{2}$ .

We proceed to higher dimensional solvable subalgebras, one dimension at a time, building from the previous dimension. Each dimension is further subdivided according to the nature of the Jordan decomposition of its generators, much as in the one-dimensional case. When advantageous–specifically in dimensions three through five–we further differentiate elements that are regular and non-regular. Following this general strategy, we get the classification of solvable subalgebras in Tables 1-5.

To assist in the identification of the solvable subalgebras (with respect to the partial classification of solvable Lie algebras described in [ŠW14]) found in the classification, we calculate certain data for each subalgebra. We often do this with the assistance of the computer algebra system Maple.

The following is a list of useful data or characteristics that may be computed with Maple: indecomposability; computation of the nilradical; computation of a basis for the Lie algebra of derivations; and the dimensions of the subalgebras in the derived series, upper central series, and lower central series. The aforementioned data may not uniquely identify the subalgebra in question, but it will narrow down the possibilities. The generalized Casimir invariants and cohomology theory may also be employed to aid with identification.

2.2. Classifying and identifying the Levi decomposable subalgebras of $C_{2}$

To determine the Levi decomposable subalgebras of $C_{2}$ , we first decompose $C_{2}$ with respect to the adjoint actions of each of its semisimple subalgebras. For instance, decomposing $C_{2}$ with respect to the simple subalgebra $\langle X_{\alpha+2\beta},$ $Y_{\alpha+2\beta},T_{1,0}\rangle$ $\cong A_{1}$ yields

[TABLE]

where $V(n)$ is the $n+1$ dimensional, irreducible representation of $A_{1}$ . In the decomposition (the first instance of) $V(2)$ is isomorphic to $A_{1}$ as a subalgebra. The remaining representations in the decomposition, or combinations thereof, give us the potential radical components for Levi decomposable subalgebras of $C_{2}$ , having $\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle$ as the Levi factor.

We find that the following is a complete list of inequivalent subalgebras of $C_{2}$ having $\langle X_{\alpha+2\beta},Y_{\alpha+2\beta},T_{1,0}\rangle$ as the Levi factor:

[TABLE]

We continue this procedure for each semisimple subalgebra of $C_{2}$ to obtain a classification of the Levi decomposable subalgebras of $C_{2}$ .

To identify the Levi decomposable subalgebras with respect to the (partial) classification of Levi decomposable subalgebra described in [ŠW14], we identify the radical, and the representation of the Levi factor on the radical. In this manner, we get the following identification of the subalgebras in Eq. (11):

[TABLE]

Precise descriptions of the Levi decomposable algebras listed above may be found in the appendices.

3. The subalgebras of $A_{2}$

The simple Lie algebra $A_{2}$ may be realized as the special linear algebra $\mathfrak{sl}(3,\mathbb{C})$ of traceless $3\times 3$ matrices with complex entries. The root system associated with $A_{2}$ is depicted in Figure 2.

It has positive roots $\alpha,~{}\beta$ , and $\alpha+\beta$ . Under the identification with $\mathfrak{sl}(3,\mathbb{C})$ , $A_{2}$ has a Chevalley basis

[TABLE]

where

[TABLE]

$H_{\alpha}$ and $H_{\beta}$ are the coroots; $X_{\alpha},X_{\beta}$ , and $X_{\alpha+\beta}$ are the positive root vectors; and $Y_{\alpha},Y_{\beta}$ , and $Y_{\alpha+\beta}$ are the negative root vectors.

We highlight three distinguished subalgebras: A Cartan subalgebra $\mathfrak{h}=\langle H_{\alpha},H_{\beta}\rangle$ ; a Borel subalgebra $\mathfrak{b}=\langle H_{\alpha},H_{\beta},X_{\alpha},X_{\beta},X_{\alpha+\beta}\rangle$ ; and the nilradical of the Borel subalgebra $\mathfrak{n}=\langle X_{\alpha},X_{\beta},X_{\alpha+\beta}\rangle$ .

The Lie group corresponding to $A_{2}$ , under its identification with $\mathfrak{sl}(3,\mathbb{C})$ , is the special linear group $\mathrm{SL}(3,\mathbb{C})$ . It is the Lie group of $3\times 3$ matrices with complex entries and determinant $1$ .

The classification and identification of the subalgebras of $A_{2}$ are described in the following tables. The classification is from [DR16b], and the identification was established as part of the present article. The classification is organized by dimension. Tables 7-8 contain the classification and identification of the solvable subalgebras, and Table 9 contains the classification and identification of the semisimple and Levi decomposable subalgebras.

4. The subalgebras of $A_{1}\times A_{1}$

The semisimple, and not simple, Lie algebra $A_{1}\times A_{1}$ may be realized as the special orthogonal algebra $\mathfrak{so}(4,\mathbb{C})$ of $4\times 4$ complex matrices satisfying $X^{t}=-X$ . It is isomorphic to the semisimple Lie algebra $\mathfrak{sl}(2,\mathbb{C})\oplus\mathfrak{sl}(2,\mathbb{C})$ , where $\mathfrak{sl}(2,\mathbb{C})$ is the special linear algebra of traceless $2\times 2$ matrices with complex entries. The root system associated with $A_{1}\times A_{1}$ is depicted in Figure 3.

It has positive roots $\alpha$ and $\beta$ . Under the identification with $\mathfrak{sl}(2,\mathbb{C})\oplus\mathfrak{sl}(2,\mathbb{C})$ , $A_{1}\times A_{1}$ has a Chevalley basis

[TABLE]

where

[TABLE]

$H_{\alpha}$ and $H_{\beta}$ are the coroots; $X_{\alpha}$ and $X_{\beta}$ are the positive root vectors; and $Y_{\alpha}$ and $Y_{\beta}$ are the negative root vectors.

We highlight three distinguished subalgebras: A Cartan subalgebra $\mathfrak{h}=\langle H_{\alpha},H_{\beta}\rangle$ ; a Borel subalgebra $\mathfrak{b}=\langle H_{\alpha},H_{\beta},X_{\alpha},X_{\beta}\rangle$ ; and the nilradical of the Borel subalgebra $\mathfrak{n}=\langle X_{\alpha},X_{\beta}\rangle$ .

The Lie group corresponding to $A_{1}\times A_{1}$ , under its identification with $\mathfrak{so}(4,\mathbb{C})$ , is the special orthogonal group $\mathrm{SO}(4,\mathbb{C})$ . It is the Lie group of $4\times 4$ complex matrices $M$ with determinant $1$ satisfying $M^{t}M=I$ .

The classification and identification of the subalgebras of $A_{1}\times A_{1}$ are described in the following tables. The classification is from [DR16a], and the identification was established as part of the present article. The classification is organized by dimension. Table 10 contains the classification and identification of the solvable subalgebras, and Table 11 contains the classification and identification of the semisimple and Levi decomposable subalgebras.

5. The subalgebras of $G_{2}$

The exceptional Lie algebra $G_{2}$ is the $14$ -dimensional simple Lie algebra of rank two. Its root system is depicted in Figure 4, and it has positive roots

[TABLE]

$G_{2}$ has a Chevalley basis

[TABLE]

with nonzero commutation relations [May16], where $X_{-\mu}=Y_{\mu}$ for $\mu\in\Phi^{+}$ ,

[TABLE]

and we set $(\alpha,\alpha)=1$ , $(\beta,\beta)=3$ , and $(\alpha,\beta)=-3/2$ .

$H_{\alpha}$ and $H_{\beta}$ are the coroots; $X_{\alpha},X_{\beta},X_{\alpha+\beta},X_{2\alpha+\beta},$ $X_{3\alpha+\beta}$ , and $X_{3\alpha+2\beta}$ are the positive root vectors; and $Y_{\alpha},Y_{\beta},Y_{\alpha+\beta},Y_{2\alpha+\beta},$ $Y_{3\alpha+\beta}$ , and $Y_{3\alpha+2\beta}$ are the negative root vectors.

We highlight three distinguished subalgebras: A Cartan subalgebra $\mathfrak{h}=\langle H_{\alpha},H_{\beta}\rangle$ ; a Borel subalgebra $\mathfrak{b}=\langle H_{\alpha},H_{\beta},X_{\alpha},X_{\beta},X_{\alpha+\beta},X_{2\alpha+\beta},$ $X_{3\alpha+\beta},X_{3\alpha+2\beta}\rangle$ ; and the nilradical of the Borel subalgebra $\mathfrak{n}=\langle X_{\alpha},X_{\beta},$ $X_{\alpha+\beta},X_{2\alpha+\beta},$ $X_{3\alpha+\beta},X_{3\alpha+2\beta}\rangle$ .

The classification and identification of the subalgebras of $G_{2}$ are described in the following tables. The classification is from [May16], and the identification was established as part of the present article. The classification is organized by dimension, and further divided into regular and non-regular subalgebras. Tables 12-20 contain the classification and identification of the solvable subalgebras, Table 21 contains the classification of semisimple subalgebras, and Table 22 contains the classification and identification of the Levi decomposable subalgebras.

Acknowledgements

The work of A.D. is partially supported by a research grant from the Professional Staff Congress/City University of New York (Grant No. TRADA-47-36). The work of J.R. is partially supported by the Natural Sciences and Engineering Research Council (Grant No. 3166-09). We also greatly thank Matthew Wu who checked our computations in the $G_{2}$ case.

Appendix A Classification of Lie algebras

By Levi’s Theorem [[ŠW14], Chapter II, Section 2], a complex Lie algebra is either semisimple, solvable, or a nontrivial semidirect sum of the first two. A Lie algebra that is a nontrivial semidirect sum of a semisimple algebra with a solvable algebra is called a Levi decomposable algebra. Hence, the classification of Lie algebras may be divided into three cases: Semisimple Lie algebras, solvable Lie algebras, and Levi decomposable algebras.

A.1. Classification of semisimple Lie algebras

Semsimple Lie algebras over $\mathbb{C}$ are direct sums of simple Lie algebras. Simple Lie algebras over $\mathbb{C}$ were classified by Wilhelm Killing [Kil88-90] and Élie Cartan [Car33]. Killing first described the classification, and Cartan provided a rigorous proof.

Under the classification, there are four families of classical Lie algebras, and five exceptional Lie algebras. The four families of classical Lie algebras are denoted $A_{n}$ ( $n\geq 1$ ), $B_{n}$ ( $n\geq 3$ ), $C_{n}$ ( $n\geq 2$ ), and $D_{n}$ ( $n\geq 4$ ). The exceptional Lie algebras are denoted $E_{6},E_{7}$ , $E_{8}$ , $F_{4}$ , and $G_{2}$ . The subscript indicates the rank of the Lie algebra. A complete description of the classification and its proof can found in any standard reference on Lie algebras such as Humphreys [Hum72].

A.2. Classification of solvable Lie algebras

A full classification of solvable Lie algebras is not known and thought to be an impossible task. However, partial classifications of solvable Lie algebras do exist. Two such partial classifications are that described by Šnobl and Winternitz in [ŠW14] and that of de Graaf [dGr05]. The classification of solvable subalgebras in this article are described, where possible, with respect to the classification described by Šnobl and Winternitz.

The classification of solvable Lie algebras presented by Šnobl and Winternitz in [ŠW14] is up to and including dimension $6$ and includes only Lie algebras indecomposable over $\mathbb{R}$ . This classification is an amalgam of results from various sources (e.g., [Bia98, Kru54, Lie88, Mor58, PSWZ76, ST13, Tur88, Tur90]).

We present the classification from [ŠW14] in its entirety up to and including dimension $3$ . Below we give a partial description of the classification in dimensions $4$ , $5$ and $6$ , including just those algebras which appear in this article. In this classification, an algebra designated with $\mathfrak{n}$ is nilpotent, and one with $\mathfrak{s}$ is solvable, but not nilpotent. The first subscript indicates the dimension, and the second index is for enumeration. So, $\mathfrak{s}_{6,242}$ is the $242^{\text{nd}}$ six-dimensional, solvable, non-nilpotent Lie algebra in the classification.

[TABLE]

We provide multiplication tables for the remaining solvable Lie algebras which appear in this article. These solvable Lie algebras do not appear in [ŠW14], but we have chosen notation consistent with that in [ŠW14]. For these solvable Lie algebras, we also identify their nilradical.

•

$\mathfrak{s}_{7,1}$ , nilradical $\mathfrak{n}_{5,2}$

[TABLE]

•

$\mathfrak{s}_{7,2}$ , nilradical $\mathfrak{n}_{5,3}$

[TABLE]

•

$\mathfrak{s}_{7,3}$ , nilradical $\mathfrak{n}_{6,18}$ , where $\mathfrak{s}_{7,3,A,B}\cong\mathfrak{s}_{7,3,A^{\prime},B^{\prime}}$ iff $AB^{\prime}=A^{\prime}B$

[TABLE]

•

$\mathfrak{s}_{8,1}$ , nilradical $\mathfrak{n}_{6,18}$

[TABLE]

A.3. Classification of Levi decomposable algebras

The Levi decomposable algebras up to dimension $9$ were classified by Turkowski [Tur88, Tur90], with some small omissions identified by Campoamor-Strusberg [CS09]. A detailed description of the classification, and general properties of Levi decomposable algebras is contained in Šnobl and Winternitz [ŠW14].

Recall that a Levi decomposable algebra $\mathfrak{g}$ is the semidirect sum of a semisimple Lie algebra $\mathfrak{p}$ , called the Levi factor, and the radical $\mathfrak{r}$ :

[TABLE]

An important observation is that $\text{ad}(\mathfrak{p})|_{\mathfrak{r}}$ is a representation of $\mathfrak{p}$ on $\mathfrak{r}$ .

The Levi factor of each Levi decomposable subalgebra within this article is $A_{1}$ . It has basis $\{h,p_{+},p_{-}\}$ with commutation relations

[TABLE]

For each natural number $j$ , $A_{1}$ has an irreducible representation of dimension $j$ denoted $\rho_{j}:A_{1}\rightarrow\mathfrak{gl}(V)$ , given, relative to a basis $\{v_{0},v_{1},...,$ $v_{j-1}\}$ of $V$ , by

[TABLE]

Below, we describe just those Levi decomposable algebras that are indecomposable and appear in this article, using the notation of [ŠW14]. For each such Levi decomposable algebra, we include a full multiplication table, identify its radical, and describe the representation $\rho$ of the Levi factor on its radical. Not all Levi decomposable algebras below appear in [ŠW14], but in cases where they don’t occur, we choose notation consistent with that in [ŠW14].

•

$A_{1}\inplus 2\mathfrak{n}_{1,1}$ , $\rho=\rho_{2}$

[TABLE]

•

$A_{1}\inplus 3\mathfrak{n}_{1,1}$ , $\rho=\rho_{3}$

[TABLE]

•

$A_{1}\inplus\mathfrak{s}_{3,1,A=1}$ , $\rho=\rho_{2}\oplus\rho_{1}$

[TABLE]

•

$A_{1}\inplus\mathfrak{s}_{4,3,A=B=1}$ , $\rho=\rho_{3}\oplus\rho_{1}$

[TABLE]

•

$A_{1}\inplus\mathfrak{s}_{4,3,A=1,B=2/3}$ , $\rho=\rho_{2}\oplus\rho_{1}\oplus\rho_{1}$

[TABLE]

•

$A_{1}\inplus\mathfrak{n}_{3,1}$ , $\rho=\rho_{2}\oplus\rho_{1}$

[TABLE]

•

$A_{1}\inplus\mathfrak{s}_{4,8,A=1}$ , $\rho=\rho_{2}\oplus\rho_{1}\oplus\rho_{1}$

[TABLE]

•

$A_{1}\inplus\mathfrak{n}_{5,3}$ , $\rho=\rho_{4}\oplus\rho_{1}$

[TABLE]

•

$A_{1}\inplus\mathfrak{n}_{5,2}$ , $\rho=2\rho_{2}\oplus\rho_{1}$

[TABLE]

•

$A_{1}\inplus\mathfrak{s}_{6,168,A=B=0}$ , $\rho=2\rho_{1}\oplus\rho_{4}$

[TABLE]

•

$A_{1}\inplus\mathfrak{s}_{6,155,A=1}$ , $\rho=2\rho_{1}\oplus 2\rho_{2}$

[TABLE]

Bibliography32

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[Bia 98] Bianchi, L.,“Sugli spazii a tre dimensioni che ammettono un grouppo continuo di movimenti,” Soc. Ital. Sci. Mem. di Mat. 11 (1898), 267-352.
2[CS 09] Campamor-Strusberg, R.: Structural data and invariants of nine-dimensional real Lie algebras with nontrivial Levi decomposition. Nova Science Publishers Inc., New York, 2009.
3[Car 33] Cartan, É.: Sur la structure des groupes des transformations finis et continus, 2nd ed., (Vuibert, Paris, 1933).
4[CM 93] Collingwood, D.H. and Mc Govern, W.M.: Nilpotent orbits in semisimple Lie algebras (Van Nostrand Reinhold, New York, 1983).
5[DKLW 86] David, D., Kamran, N., Levi, D., and Winternitz ,P., “Symmetry reduction for the Kadomtsev-Petviashvili equation using a loop algebra,” J. Math. Phys. 27 1225, (1986).
6[d Gr 05] de Graaf, W.A. “Classification of solvable Lie algebras,” Experiment. Math. Volume 14, Issue 1, 15-25 (2005).
7[d Gr 09] de Graaf, W.A.: SLA-computing with Simple Lie algebras. a GAP package , (2009). http://www.science.unitn.it/degraaf/sla.html
8[d Gr 11] de Graaf, W.A., “Constructing semisimple subalgebras of semisimple Lie algebras,” J. Algebra, 325, 416-430 (2011).

TL;DR

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Subalgebras of the rank two semisimple Lie algebras

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

Contents

List of Tables

1. Introduction

2. The subalgebras of C2C_{2}C2​

2.1. Classifying and identifying the solvable subalgebras of C2C_{2}C2​

2.2. Classifying and identifying the Levi decomposable subalgebras of C2C_{2}C2​

3. The subalgebras of A2A_{2}A2​

4. The subalgebras of A1×A1A_{1}\times A_{1}A1​×A1​

5. The subalgebras of G2G_{2}G2​

Acknowledgements

Appendix A Classification of Lie algebras

A.1. Classification of semisimple Lie algebras

A.2. Classification of solvable Lie algebras

A.3. Classification of Levi decomposable algebras

2. The subalgebras of $C_{2}$

2.1. Classifying and identifying the solvable subalgebras of $C_{2}$

2.2. Classifying and identifying the Levi decomposable subalgebras of $C_{2}$

3. The subalgebras of $A_{2}$

4. The subalgebras of $A_{1}\times A_{1}$

5. The subalgebras of $G_{2}$