On the compact real forms of the Lie algebras of type $E_6$ and $F_4$

TL;DR

This paper constructs the compact real forms of Lie algebras of types E6 and F4, using subgroup structures and elementary proofs to establish their existence and properties.

Contribution

It provides a new construction of the compact real form of E6 using a specific finite subgroup and offers an elementary proof of its Lie algebra structure.

Findings

Constructed the compact real form of E6 via subgroup methods

Proved the Lie algebra satisfies the Jacobi identity from first principles

Identified the F4 algebra as a subalgebra

Abstract

We give a construction of the compact real form of the Lie algebra of type $E_6$, using the finite irreducible subgroup of shape $3^{3+3}:\mathrm{SL}_3(3)$, which is isomorphic to a maximal subgroup of the orthogonal group $\Omega_7(3)$. In particular we show that the algebra is uniquely determined by this subgroup. Conversely, we prove from first principles that the algebra satisfies the Jacobi identity, and thus give an elementary proof of existence of a Lie algebra of type $E_6$. The compact real form of $F_4$ is exhibited as a subalgebra.