Non-naturally reductive Einstein metrics on $\mathrm{SO}(n)$

TL;DR

This paper proves that for all sufficiently large n, the special orthogonal group SO(n) admits multiple distinct non-naturally reductive Einstein metrics, expanding understanding of geometric structures on Lie groups.

Contribution

It establishes the existence of numerous non-naturally reductive Einstein metrics on SO(n) for n ≥ 10, providing new examples and advancing the classification of Einstein metrics on Lie groups.

Findings

At least 2([ (n-1)/3 ] - 2) such metrics exist for each n ≥ 10

The metrics are non-naturally reductive and left-invariant on SO(n)

The result applies to all sufficiently large n, specifically n ≥ 10

Abstract

In this article, we prove that every compact simple Lie group $SO(n)$ for $n\geq 10$ admits at least $2\left([\frac{n-1}{3}]-2\right)$ non-naturally reductive left-invariant Einstein metrics.

Equations

$$\mathfrak{m}={\mathfrak{m}}_1\oplus \cdots \oplus {\mathfrak{m}}_q$$

$$⟨\cdot ,\cdot ⟩=x_1(-B){|}_{{\mathfrak{m}}_1}+\cdots +x_q(-B){|}_{{\mathfrak{m}}_q},$$

$$⟨[X,Y{]}_{\mathfrak{m}},Z⟩+⟨Y,[X,Z{]}_{\mathfrak{m}}⟩=0,\forall X,Y,Z\in \mathfrak{m}.$$

$$r=y_1(-B){|}_{{\mathfrak{m}}_1}+\cdots +y_q(-B){|}_{{\mathfrak{m}}_q},$$

$$(ijk):=\sum (A_{\alpha ,\beta }^{\gamma }{)}^2,$$

$$r_k=\frac{1}{2x_k}+\frac{1}{4d_k}\sum _{j,i}\frac{x_k}{x_j{x}_i}(kji)-\frac{1}{2d_k}\sum _{j,i}\frac{x_j}{x_k{x}_i}(jki),(k=0,1,\cdots ,p+q).$$

$$\mathrm{S}\mathrm{O}(k_1)\times \mathrm{S}\mathrm{O}(k_2)\times \mathrm{S}\mathrm{O}(k_3)\to \mathrm{S}\mathrm{O}(k_1+k_2+k_3)\to \mathrm{S}\mathrm{O}(k_1+k_2+k_3)/(\mathrm{S}\mathrm{O}(k_1)\times \mathrm{S}\mathrm{O}(k_2)\times \mathrm{S}\mathrm{O}(k_3)).$$

$$\mathfrak{s}\mathfrak{o}(k_1+k_2+k_3)=\mathfrak{s}\mathfrak{o}(k_1)\oplus \mathfrak{s}\mathfrak{o}(k_2)\oplus \mathfrak{s}\mathfrak{o}(k_3)\oplus \mathfrak{m}.$$

$$\mathfrak{s}\mathfrak{o}(k_1+k_2+k_3)={\mathfrak{m}}_1\oplus {\mathfrak{m}}_2\oplus {\mathfrak{m}}_3\oplus {\mathfrak{m}}_{12}\oplus {\mathfrak{m}}_{13}\oplus {\mathfrak{m}}_{23},$$

$$G/\{e\}\cong (G\times \mathrm{S}\mathrm{O}(k_1)\times \mathrm{S}\mathrm{O}(k_2)\times \mathrm{S}\mathrm{O}(k_3))/\mathrm{diag}(\mathrm{S}\mathrm{O}(k_1)\times \mathrm{S}\mathrm{O}(k_2)\times \mathrm{S}\mathrm{O}(k_3)).$$

$$\begin{array}{rl}⟨\cdot ,\cdot ⟩& =x_1(-B){|}_{{\mathfrak{m}}_1}+x_2(-B){|}_{{\mathfrak{m}}_2}+x_3(-B){|}_{{\mathfrak{m}}_3}\\ & +x_{12}(-B){|}_{{\mathfrak{m}}_{12}}+x_{13}(-B){|}_{{\mathfrak{m}}_{13}}+x_{23}(-B){|}_{{\mathfrak{m}}_{23}}.\end{array}$$

$$\begin{array}{rl}{r}_1& =\frac{k_1-2}{4(n-2)x_1}+\frac{1}{4(n-2)}\left(k_2\frac{x_1}{x_{12}^2}+k_3\frac{x_1}{x_{13}^2}\right),\\ r_2& =\frac{k_2-2}{4(n-2)x_2}+\frac{1}{4(n-2)}\left(k_1\frac{x_2}{x_{12}^2}+k_3\frac{x_2}{x_{23}^2}\right),\\ r_3& =\frac{k_3-2}{4(n-2)x_3}+\frac{1}{4(n-2)}\left(k_1\frac{x_3}{x_{13}^2}+k_2\frac{x_3}{x_{23}^2}\right),\\ r_{12}& =\frac{1}{2x_{12}}+\frac{k_3}{4(n-2)}\left(\frac{x_{12}}{x_{13}{x}_{23}}-\frac{x_{13}}{x_{12}{x}_{23}}-\frac{x_{23}}{x_{12}{x}_{13}}\right)-\frac{1}{4(n-2)}\left(\frac{(k_1-1)x_1}{x_{12}^2}+\frac{(k_2-1)x_2}{x_{12}^2}\right),\\ r_{13}& =\frac{1}{2x_{13}}+\frac{k_2}{4(n-2)}\left(\frac{x_{13}}{x_{12}{x}_{23}}-\frac{x_{12}}{x_{13}{x}_{23}}-\frac{x_{23}}{x_{12}{x}_{13}}\right)-\frac{1}{4(n-2)}\left(\frac{(k_1-1)x_1}{x_{13}^2}+\frac{(k_3-1)x_3}{x_{13}^2}\right),\\ r_{23}& =\frac{1}{2x_{23}}+\frac{k_1}{4(n-2)}\left(\frac{x_{23}}{x_{13}{x}_{12}}-\frac{x_{13}}{x_{12}{x}_{23}}-\frac{x_{12}}{x_{23}{x}_{13}}\right)-\frac{1}{4(n-2)}\left(\frac{(k_2-1)x_2}{x_{23}^2}+\frac{(k_3-1)x_3}{x_{23}^2}\right),\end{array}$$

$$⟨[X,Y{]}_{\mathfrak{m}},Z⟩+⟨Y,[X,Z{]}_{\mathfrak{m}}⟩=0,\forall X,Y,Z\in \mathfrak{m}.$$

$$r_1=r_2,r_2=r_3,r_3=r_{12},r_{12}=r_{13},r_{13}=r_{23}$$

$$\{\begin{array}{rl}{f}_1& =-2k{x}_2{x_3}^2{x_{12}}^2+l{x_2}^2{x}_3{x_{12}}^2+k{x_2}^2{x}_3+k{x}_3{x_{12}}^2-l{x}_2{x_{12}}^2+2x_2{x_{12}}^2-2x_3{x_{12}}^2=0,\\ f_2& =l{x_2}^2{x_{12}}^2-l{x}_2{x_{12}}^3+3k{x_2}^2-4x_2{x}_{12}k+{x_{12}}^{2}k-2{x_2}^2+4x_2{x}_{12}-2{x_{12}}^2=0,\\ f_3& =k{x}_2{x}_3+2k{x_3}^2+k{x}_3{x}_{12}+l{x_3}^2-4k{x}_3-2x_{3}l-x_2{x}_3-{x_3}^2+l+4x_3-2=0.\end{array}$$

$$f_2=\left(x_2-x_{12}\right)\left(l{x}_2{x_{12}}^2+3k{x}_2-k{x}_{12}-2x_2+2x_{12}\right).$$

$$x_2=\frac{x_{12}\left(k-2\right)}{l{x_{12}}^2+3k-2}.$$

$$\{\begin{array}{rl}{g}_1=& 2k^{2}l{x_3}^2{x_{12}}^3-k{l}^2{x}_3{x_{12}}^4-4kl{x_3}^2{x_{12}}^3+2l^2{x}_3{x_{12}}^4+6k^3{x_3}^2{x}_{12}-7k^{2}l{x}_3{x_{12}}^2+k{l}^2{x_{12}}^3\\ & -16k^2{x_3}^2{x}_{12}+20kl{x}_3{x_{12}}^2-2kl{x_{12}}^3-2l^2{x_{12}}^3-10k^3{x}_3+3k^{2}l{x}_{12}+8k{x_3}^2{x}_{12}\\ & -12l{x}_3{x_{12}}^2+4l{x_{12}}^3+34k^2{x}_3-6k^2{x}_{12}-8kl{x}_{12}-32k{x}_3+16k{x}_{12}+4x_{12}l+8x_3\\ & -8x_{12}=0,\\ g_2=& 2kl{x_3}^2{x_{12}}^2+kl{x}_3{x_{12}}^3+l^2{x_3}^2{x_{12}}^2-4kl{x}_3{x_{12}}^2-2l^2{x}_3{x_{12}}^2-l{x_3}^2{x_{12}}^2+6k^2{x_3}^2\\ & +4k^2{x}_3{x}_{12}+3kl{x_3}^2+l^2{x_{12}}^2+4l{x}_3{x_{12}}^2-12k^2{x}_3-6kl{x}_3-7k{x_3}^2-5k{x}_3{x}_{12}-2l{x_3}^2\\ & -2l{x_{12}}^2+3kl+20k{x}_3+4x_{3}l+2{x_3}^2+2x_{12}{x}_3-6k-2l-8x_3+4=0\end{array}$$

$$\begin{array}{rl}& h(0)=4(5k-2{)}^2(k-1{)}^2(l-1+2k),\\ & h(1)=(k-1)(l-1+2k)(k-l)(2k+l{)}^2\\ & h(+\infty )\to +\infty .\end{array}$$

$$\{x_1=x_2=\frac{x_{12}\left(k-2\right)}{l{x_{12}}^2+3k-2},x_3=\alpha (x_{12}),x_{12}\ne 1,x_{13}=x_{23}=1\},$$

Full text

1. Introduction

A Riemannian manifold $(M,\langle\cdot,\cdot\rangle)$ is called Einstein if there exists a constant $\lambda$ such that $\mathrm{Ric}=\lambda\langle\cdot,\cdot\rangle$, where $\mathrm{Ric}$ is the Ricci tensor of the metric $\langle\cdot,\cdot\rangle$ . For a survey of the results on Einstein metrics, we refer to the book [3] and the papers [14, 15]. For the study on Einstein metrics of homogeneous manifolds, see [4, 5, 16].

There are a lot of important results on Einstein metrics on Lie groups which are a special class of homogeneous manifolds. It is shown in [11] that any Einstein solvmanifold, i.e. a simply connected solvable Lie group admitting a left-invariant metric, is standard. According to a conjecture of Alekseevskii ([3]), this exhausts noncompact Lie groups with left-invariant Einstein metrics. Also Einstein metrics on solvable Lie groups are unique to isometry and scaling [10], which is in sharp contrast to the compact setting.

In [9], D’ Atri and Ziller prove every compact simple Lie group except ${\mathrm{S}}{\mathrm{O}}(3)$ admits at least two left-invariant Einstein metrics which are all naturally reductive. Meanwhile, they pose the question whether there exist non-naturally reductive left-invariant Einstein metrics on compact simple Lie groups. From then on, there are a lot of studies on non-naturally reductive Einstein metrics on compact simple Lie groups. In [12], Mori obtains non-naturally reductive left-invariant Einstein metrics on ${\mathrm{S}}{\mathrm{U}}(n)$ for $n\geq 6$. Then in [1], the authors prove the existence of left-invariant Einstein metrics on compact Lie groups ${\mathrm{S}}{\mathrm{O}}(n)$ for $n\geq 11$, ${\mathrm{S}}{\mathrm{p}}(n)$ for $n\geq 3$, $\mathrm{E}_{6},\mathrm{E}_{7}$ and $\mathrm{E}_{8}$. After that, Chen and Liang [7] give one non-naturally reductive left-invariant Einstein metric on $\mathrm{F}_{4}$. In [2], Arvanitoyeorgos, Sakane and Statha obtain left-invariant Einstein metrics on compact Lie groups ${\mathrm{S}}{\mathrm{O}}(n)$ for $n\geq 7$, which are not naturally reductive. Recently, Chrysikos and Sakane find new non-naturally reductive Einstein metrics on exceptional Lie groups in [8], especially they give the first non-naturally reductive Einstein metric on ${\mathrm{G}}_{2}$.

That is to say, every compact simple Lie group except some small groups admits non-naturally reductive left-invariant Einstein metrics. But how many non-naturally reductive left-invariant Einstein metrics does every compact simple Lie group possibly admit? It is natural to discuss the lower bound of the number of non-naturally reductive left-invariant Einstein metrics on compact simple Lie groups, especially classical Lie groups.

Yan and Deng prove an interesting result in [17]: for any integer $n=p_{1}^{l_{1}}p_{2}^{l_{2}}\cdots p_{s}^{l_{s}}$ with $p_{i}$ prime and $p_{i}\not=p_{j}$,

1. (1)

${\mathrm{S}}{\mathrm{O}}(2n)$ admits at least $(l_{1}+1)(l_{2}+1)\cdots(l_{s}+1)-3$ non-equivalent non-naturally reductive Einstein metrics, 2. (2)

and ${\mathrm{S}}{\mathrm{p}}(2n)$ admits at least $(l_{1}+1)(l_{2}+1)\cdots(l_{s}+1)-1$ non-equivalent non-naturally reductive Einstein metrics.

That is, they give lower bounds of the number for ${\mathrm{S}}{\mathrm{O}}(2n)$ and ${\mathrm{S}}{\mathrm{p}}(2n)$, which depend on the integer $n$. But we point out that the above results hold only when $n$ is not a prime. A much better estimate for ${\mathrm{S}}{\mathrm{p}}(n)$ is given in [6] that every ${\mathrm{S}}{\mathrm{p}}(n)$ admits at least $2[\frac{n-1}{3}]$ non-naturally reductive left-invariant Einstein metrics for $n\geq 4$.

In this article, we obtain the following lower bound of the number of non-naturally reductive left-invariant Einstein metrics on ${\mathrm{S}}{\mathrm{O}}(n)$.

Theorem 1.1**.**

For every integer $n\geq 10$, ${\mathrm{S}}{\mathrm{O}}(n)$ admits at least $2\left([\frac{n-1}{3}]-2\right)$ non-naturally reductive left-invariant Einstein metrics.

The paper is organized as follows. In section 2, we recall the study on non-naturally reductive left-invariant metrics on ${\mathrm{S}}{\mathrm{O}}(n)$ in [2], in particular the Ricci tensor of a class of left-invariant metrics $\langle\cdot,\cdot\rangle$ on ${\mathrm{S}}{\mathrm{O}}(n)$, and sufficient and necessary conditions for $\langle\cdot,\cdot\rangle$ to be naturally reductive. Furthermore, they prove that ${\mathrm{S}}{\mathrm{O}}(n)$ admits non-naturally reductive Einstein metrics which are ${\mathrm{A}}{\mathrm{d}}({\mathrm{S}}{\mathrm{O}}(n-6)\times{\mathrm{S}}{\mathrm{O}}(3)\times{\mathrm{S}}{\mathrm{O}}(3))$-invariant. In section 3, based on the Ricci tensor formulae in [2] and the technique of Gröbner basis, we prove that ${\mathrm{S}}{\mathrm{O}}(2k+l)$ admits at least two non-naturally reductive Einstein metrics which are ${\mathrm{A}}{\mathrm{d}}({\mathrm{S}}{\mathrm{O}}(k)\times{\mathrm{S}}{\mathrm{O}}(k)\times{\mathrm{S}}{\mathrm{O}}(l))$-invariant when $l>k\geq 3$. It implies Theorem 1.1.

2. The study on ${\mathrm{S}}{\mathrm{O}}(n)$ in [2]

Let $G$ be a compact semisimple Lie group and let $K$ be a connected closed subgroup of $G$ with Lie algebras $\mathfrak{g}$ and $\mathfrak{k}$. Since the Killing form $B$ of $\mathfrak{g}$ is negative definite, $-B$ is an ${\mathrm{A}}{\mathrm{d}}(G)$-invariant inner product on $\mathfrak{g}$. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}$ be the reductive decomposition of $\mathfrak{g}$ such that $[\mathfrak{k},\mathfrak{m}]\subset\mathfrak{m}$. Then $\mathfrak{m}$ can be identified with the tangent space of $G/K$ at the origin. Assume that $\mathfrak{m}$ admits a decomposition into mutually non-equivalent irreducible ${\mathrm{A}}{\mathrm{d}}(K)$-modules:

[TABLE]

Then any $G$-invariant metric on $G/K$ has the following form

[TABLE]

where $x_{1},\cdots,x_{q}\in\mathbb{R}^{+}$. The $G$-invariant metric $\langle\cdot,\cdot\rangle$ on $G/K$ is called naturally reductive if

[TABLE]

Note that $G$-invariant symmetric covariant $2$-tensors on $G/K$ are of the same form as Riemannian metrics. In particular, the Ricci tensor $r$ of a $G$-invariant Riemannian metric on $G/K$ is of the same form as (2.2), that is,

[TABLE]

where $y_{1},\cdots,y_{q}\in\mathbb{R}$. Let $d_{i}=\dim\mathfrak{m}_{i}$ and let $\{e^{i}_{\alpha}\}_{\alpha=1}^{d_{i}}$ be a $(-B)$-orthonormal basis of $\mathfrak{m}_{i}$. Denote $A_{\alpha,\beta}^{\gamma}=-B([e_{\alpha}^{i},e_{\beta}^{j}],e_{\gamma}^{k})$, i.e. $[e_{\alpha}^{i},e_{\beta}^{j}]=\sum_{\gamma}A_{\alpha,\beta}^{\gamma}e_{\gamma}^{k}$, and define

[TABLE]

where the sum is taken over all indices $\alpha,\beta,\gamma$ with $e_{\alpha}^{i}\in\mathfrak{m}_{i},e_{\beta}^{j}\in\mathfrak{m}_{j},e_{\gamma}^{k}\in\mathfrak{m}_{k}$. Then $(ijk)$ is independent of the choice for the $-B$-orthonormal basis of $\mathfrak{m}_{i},\mathfrak{m}_{j},\mathfrak{m}_{k}$, and $(ijk)=(jik)=(jki)$. Then we have the following result.

Lemma 2.1** ([13]).**

The components $r_{1},\cdots,r_{p+q}$ of the Ricci tensor $r$ associated to $\langle\cdot,\cdot\rangle$ of the form (2.2) on $G/K$

[TABLE]

Here, the sums are taken over all $i=1,\cdots,p+q$.

For $G={\mathrm{S}}{\mathrm{O}}(k_{1}+k_{2}+k_{3})$, we consider the closed subgroup $K={\mathrm{S}}{\mathrm{O}}(k_{1})\times{\mathrm{S}}{\mathrm{O}}(k_{2})\times{\mathrm{S}}{\mathrm{O}}(k_{3})$, where the embedding of $K$ in $G$ is diagonal, and we have the fibration

[TABLE]

Denote the tangent space of ${\mathrm{S}}{\mathrm{O}}(k_{1}+k_{2}+k_{3})$, ${\mathrm{S}}{\mathrm{O}}(k_{1}+k_{2}+k_{3})/({\mathrm{S}}{\mathrm{O}}(k_{1})\times{\mathrm{S}}{\mathrm{O}}(k_{2})\times{\mathrm{S}}{\mathrm{O}}(k_{3}))$ and ${\mathrm{S}}{\mathrm{O}}(k_{1})\times{\mathrm{S}}{\mathrm{O}}(k_{2})\times{\mathrm{S}}{\mathrm{O}}(k_{3})$ at the origin by ${\mathfrak{s}}{\mathfrak{o}}(k_{1}+k_{2}+k_{3})$, ${\mathfrak{m}}$ and ${\mathfrak{s}}{\mathfrak{o}}(k_{1})\oplus{\mathfrak{s}}{\mathfrak{o}}(k_{2})\oplus{\mathfrak{s}}{\mathfrak{o}}(k_{3})$, respectively. Then we have

[TABLE]

By setting ${\mathfrak{m}}_{1}={\mathfrak{s}}{\mathfrak{o}}(k_{1}),{\mathfrak{m}}_{2}={\mathfrak{s}}{\mathfrak{o}}(k_{2})$ and ${\mathfrak{m}}_{3}={\mathfrak{s}}{\mathfrak{o}}(k_{3})$, we have the following decomposition:

[TABLE]

where ${\mathfrak{m}}_{12},{\mathfrak{m}}_{13}$ and ${\mathfrak{m}}_{23}$ are irreducible submodules of ${\mathfrak{m}}$. Note that there is a diffeomorphism

[TABLE]

Consider left-invariant metrics on $G$ which are determined by ${\mathrm{A}}{\mathrm{d}}(K)$-invariant inner products on ${\mathfrak{s}}{\mathfrak{o}}(k_{1}+k_{2}+k_{3})$ given by

[TABLE]

By Lemma 2.1, we have the following formulae of the Ricci tensor.

Lemma 2.2** ([2]).**

The components of the Ricci tensor $r$ of left-invariant metrics on $G$ defined by $\langle\cdot,\cdot\rangle$ of the form $(\ref{met1})$ are given as follows:

[TABLE]

where $n=k_{1}+k_{2}+k_{3}$.

Definition 2.3**.**

A Riemannian homogeneous space $(M=G/K,\langle\cdot,\cdot\rangle)$ with a reductive complement ${\mathfrak{m}}$ of $\mathfrak{k}$ in $\mathfrak{g}$ is called naturally reductive if

[TABLE]

Based on the criterion for a left-invariant metric on a Lie group to be naturally reductive given in [9], we have the following lemma.

Lemma 2.4** ([2]).**

If a left-invariant metric $\langle\cdot,\cdot\rangle$ of the form $(\ref{met1})$ on $G$ is naturally reductive with respect to $G\times L$, where $L$ is a closed subgroup of $G={\mathrm{S}}{\mathrm{O}}(k_{1}+k_{2}+k_{3})$, then one of the following holds:

1. (1)

$x_{1}=x_{2}=x_{12}$, $x_{13}=x_{23};$ 2. (2)

$x_{2}=x_{3}=x_{23}$, $x_{12}=x_{13};$ 3. (3)

$x_{1}=x_{3}=x_{13}$, $x_{12}=x_{23};$ 4. (4)

$x_{12}=x_{13}=x_{23}$.

Conversely, if one of the above conditions is satisfied, then there exists a closed subgroup $L$ of $G$ such that the metric $\langle\cdot,\cdot\rangle$ of the form $(\ref{met1})$ is naturally reductive with respect to $G\times L$.

3. The proof of Theorem 1.1

In [2], the authors prove that, for any $n\geq 9$, the Lie group ${\mathrm{S}}{\mathrm{O}}(n)$ admits at least one left-invariant Einstein metric determined by the ${\mathrm{A}}{\mathrm{d}}({\mathrm{S}}{\mathrm{O}}(n-6)\times{\mathrm{S}}{\mathrm{O}}(3)\times{\mathrm{S}}{\mathrm{O}}(3))$-invariant inner product of the form (2.4), which is non-naturally reductive. In this section, we first prove the following theorem.

Theorem 3.1**.**

For any $l>k\geq 3$, ${\mathrm{S}}{\mathrm{O}}(2k+l)$ admits at least two left-invariant Einstein metrics determined by ${\mathrm{A}}{\mathrm{d}}({\mathrm{S}}{\mathrm{O}}(k)\times{\mathrm{S}}{\mathrm{O}}(k)\times{\mathrm{S}}{\mathrm{O}}(l))$-invariant inner products of the form (2.4), which are non-naturally reductive.

We will prove Theorem 3.1 by solving homogeneous Einstein equations

[TABLE]

under the assumption $k=k_{1}=k_{2}$ and $l=k_{3}$. Furthermore, we consider the metric (2.4) with $x_{1}=x_{2}$. Then we have $x_{13}=x_{23}$. Standard the metric with $x_{13}=x_{23}=1$. Therefore homogeneous Einstein equations are equivalent to the following system of equations:

[TABLE]

In particular,

[TABLE]

If $x_{2}=x_{12}$, by Proposition 2.4, the left-invariant metric is naturally reductive. Assume that $x_{2}\neq x_{12}$. Then we have

[TABLE]

Substitute it into equations $f_{1}=0$ and $f_{3}=0$, we have the following equations:

[TABLE]

Consider the ideal $I$ generated by $\{g_{1},g_{2},zx_{3}x_{12}-1\}$ and the polynomial ring $R$ with coefficients in $\mathbb{Q}$, and take a lexicographic order $>$ with $z>x_{3}>x_{12}$ for a monomial ordering on $R$. By the computer software, we get two polynomials containing in the Gröbner basis for the ideal $I$.

$h(x_{12})={l}^{2}(k+l)(2\,{k}^{2}+2\,kl+{l}^{2}-l){x_{{12}}}^{8}\\ -2\,{l}^{2}(2\,k+l-2)(4\,{k}^{2}+4\,kl+{l}^{2}-l){x_{{12}}}^{7}\\ +l(16\,{k}^{4}+76\,{k}^{3}l+71\,{k}^{2}{l}^{2}+22\,k{l}^{3}+{l}^{4}-20\,{k}^{3}-119\,{k}^{2}l-81\,k{l}^{2}-16\,{l}^{3}+8\,{k}^{2}+51\,kl+19\,{l}^{2}-4\,l){x_{{12}}}^{6}\\ -4\,l(2\,k+l-2)(14\,{k}^{3}+20\,{k}^{2}l+5\,k{l}^{2}-14\,{k}^{2}-21\,kl-4\,{l}^{2}+4\,k+4\,l){x_{{12}}}^{5}\\ +(32\,{k}^{5}+344\,{k}^{4}l+368\,{k}^{3}{l}^{2}+117\,{k}^{2}{l}^{3}+6\,k{l}^{4}-80\,{k}^{4}-842\,{k}^{3}l-686\,{k}^{2}{l}^{2}-168\,k{l}^{3}-4\,{l}^{4}+82\,{k}^{3}+713\,{k}^{2}l+406\,k{l}^{2}+60\,{l}^{3}-40\,{k}^{2}-236\,kl-76\,{l}^{2}+8\,k+20\,l){x_{{12}}}^{4}\\ -2\,(2\,k+l-2)(48\,{k}^{4}+124\,{k}^{3}l+31\,{k}^{2}{l}^{2}-92\,{k}^{3}-215\,{k}^{2}l-46\,k{l}^{2}+64\,{k}^{2}+110\,kl+16\,{l}^{2}-16\,k-16\,l){x_{{12}}}^{3}\\ +(448\,{k}^{5}+608\,{k}^{4}l+212\,{k}^{3}{l}^{2}+9\,{k}^{2}{l}^{3}-1424\,{k}^{4}-1550\,{k}^{3}l-448\,{k}^{2}{l}^{2}-12\,k{l}^{3}+1714\,{k}^{3}+1427\,{k}^{2}l+304\,k{l}^{2}+4\,{l}^{3}-956\,{k}^{2}-556\,kl-64\,{l}^{2}+232\,k+76\,l-16){x_{{12}}}^{2}\\ -4\,(k-1)(5\,k-2)(-2+3\,k)(2\,k+l-2)(4\,k+l-1)x_{{12}}\\ +4\,(5\,k-2)^{2}(k-1)^{2}(l-1+2\,k)$

and

$h(x_{12},x_{3})=-{l}^{2}(k+l)({k}^{2}+4\,kl+2\,{l}^{2}-2\,k-2\,l)(2\,{k}^{2}+2\,kl+{l}^{2}-l){x_{{12}}}^{7}\\ +2\,{l}^{2}(2\,k+l-2)({k}^{2}+4\,kl+2\,{l}^{2}-2\,k-2\,l)(4\,{k}^{2}+4\,kl+{l}^{2}-l){x_{{12}}}^{6}\\ -l(16\,{k}^{6}+120\,{k}^{5}l+357\,{k}^{4}{l}^{2}+408\,{k}^{3}{l}^{3}+206\,{k}^{2}{l}^{4}+43\,k{l}^{5}+2\,{l}^{6}-52\,{k}^{5}-365\,{k}^{4}l-841\,{k}^{3}{l}^{2}-714\,{k}^{2}{l}^{3}-252\,k{l}^{4}-32\,{l}^{5}+48\,{k}^{4}+357\,{k}^{3}l+622\,{k}^{2}{l}^{2}+355\,k{l}^{3}+66\,{l}^{4}-16\,{k}^{3}-122\,{k}^{2}l-154\,k{l}^{2}-44\,{l}^{3}+8\,kl+8\,{l}^{2}){x_{{12}}}^{5}\\ +2\,l(2\,k+l-2)(28\,{k}^{5}+112\,{k}^{4}l+166\,{k}^{3}{l}^{2}+90\,{k}^{2}{l}^{3}+15\,k{l}^{4}-84\,{k}^{4}-254\,{k}^{3}l-278\,{k}^{2}{l}^{2}-114\,k{l}^{3}-14\,{l}^{4}+64\,{k}^{3}+172\,{k}^{2}l+131\,k{l}^{2}+28\,{l}^{3}-16\,{k}^{2}-32\,kl-14\,{l}^{2}){x_{{12}}}^{4}\\ +(-32\,{k}^{7}-372\,{k}^{6}l-1118\,{k}^{5}{l}^{2}-1337\,{k}^{4}{l}^{3}-710\,{k}^{3}{l}^{4}-153\,{k}^{2}{l}^{5}-7\,k{l}^{6}+144\,{k}^{6}+1664\,{k}^{5}l+3882\,{k}^{4}{l}^{2}+3620\,{k}^{3}{l}^{3}+1499\,{k}^{2}{l}^{4}+247\,k{l}^{5}+6\,{l}^{6}-242\,{k}^{5}-2641\,{k}^{4}l-4816\,{k}^{3}{l}^{2}-3415\,{k}^{2}{l}^{3}-1003\,k{l}^{4}-98\,{l}^{5}+204\,{k}^{4}+1882\,{k}^{3}l+2625\,{k}^{2}{l}^{2}+1293\,k{l}^{3}+210\,{l}^{4}-88\,{k}^{3}-588\,{k}^{2}l-586\,k{l}^{2}-150\,{l}^{3}+16\,{k}^{2}+56\,kl+32\,{l}^{2}){x_{{12}}}^{3}\\ +2\,(2\,k+l-2)(48\,{k}^{6}+156\,{k}^{5}l+263\,{k}^{4}{l}^{2}+162\,{k}^{3}{l}^{3}+27\,{k}^{2}{l}^{4}-188\,{k}^{5}-583\,{k}^{4}l-706\,{k}^{3}{l}^{2}-342\,{k}^{2}{l}^{3}-48\,k{l}^{4}+248\,{k}^{4}+688\,{k}^{3}l+631\,{k}^{2}{l}^{2}+216\,k{l}^{3}+20\,{l}^{4}-144\,{k}^{3}-316\,{k}^{2}l-216\,k{l}^{2}-40\,{l}^{3}+32\,{k}^{2}+48\,kl+20\,{l}^{2}){x_{{12}}}^{2}\\ +(-368\,{k}^{7}-740\,{k}^{6}l-850\,{k}^{5}{l}^{2}-513\,{k}^{4}{l}^{3}-120\,{k}^{3}{l}^{4}-3\,{k}^{2}{l}^{5}+1908\,{k}^{6}+3454\,{k}^{5}l+3330\,{k}^{4}{l}^{2}+1642\,{k}^{3}{l}^{3}+321\,{k}^{2}{l}^{4}+8\,k{l}^{5}-3740\,{k}^{5}-5875\,{k}^{4}l-4684\,{k}^{3}{l}^{2}-1787\,{k}^{2}{l}^{3}-256\,k{l}^{4}-4\,{l}^{5}+3546\,{k}^{4}+4678\,{k}^{3}l+2943\,{k}^{2}{l}^{2}+784\,k{l}^{3}+60\,{l}^{4}-1676\,{k}^{3}-1786\,{k}^{2}l-800\,k{l}^{2}-116\,{l}^{3}+344\,{k}^{2}+280\,kl+68\,{l}^{2}-16\,k-8\,l)x_{{12}}\\ +2\,(k-1)(k-2)(5\,k-2)(3\,{k}^{2}+2\,kl+{l}^{2}-2\,k-l)(8\,{k}^{2}+4\,kl-8\,k)\\ +2\,(l-1)(k-1)(k-2)(5\,k-2)(2\,k+l-1)(3\,{k}^{2}+(2\,k+l)(l-1))x_{3}.$

By $h(x_{12},x_{3})=0$, $x_{3}$ can be expressed by a polynomial of $x_{12}$ of degree $7$ with coefficients in $\mathbb{Q}$ for $k\geq 3$ and $l\geq 2$. It means if $x_{12}=s\in\mathbb{R}$ is a solution of $h(x_{12})=0$, then there exists $x_{3}=t\in\mathbb{R}$ such that $h(s,t)=0.$ Moreover, we have

[TABLE]

For $l>k\geq 2$, we have $h(0)>0$ and $h(1)<0$. as a result, $h(x_{12})=0$ has at least two solutions, one of which is between [math] and $1$ and the other is more than $1$. As is shown above, there exists $x_{3}\in\mathbb{R}$ with respect to each solution of $h(x_{12})=0$.

The following is to check when $x_{3}\in\mathbb{R}^{+}$. For this, take a lexicographic order $>$ with $z>x_{12}>x_{3}$ for a monomial ordering on $R$. Similarly, we have the following polynomial containing in the Gröbner basis for the ideal $I$:

$p(x_{3})=4\,(l-1+2\,k)(2\,{k}^{2}+2\,kl+l(l-1))(3\,{k}^{2}+2\,kl+{l}^{2}-2\,k-l)^{2}x_{3}^{8}\\ -16\,(2\,k+l-2)(3\,{k}^{2}+(2\,k+l)(l-1))(3\,{k}^{4}+(12\,l-2){k}^{3}+(14\,{l}^{2}-10\,l){k}^{2}+({l}^{2}(8\,l-10)+2\,l)k+2\,{l}^{3}(l-2)+2\,{l}^{2})x_{3}^{7}\\ +(160\,{k}^{7}+(1928\,l-1000){k}^{6}+(5492\,{l}^{2}-6764\,l+1408){k}^{5}+(7644\,{l}^{3}-15830\,{l}^{2}+7818\,l-736){k}^{4}+(6201\,{l}^{4}-18266\,{l}^{3}+15881\,{l}^{2}-3848\,l+128){k}^{3}+(3072\,{l}^{5}-11752\,{l}^{4}+14848\,{l}^{3}-6912\,{l}^{2}+808\,l){k}^{2}+(880\,{l}^{6}-4208\,{l}^{5}+7012\,{l}^{4}-4824\,{l}^{3}+1204\,{l}^{2}-64\,l)k+112\,{l}^{7}-664\,{l}^{6}+1432\,{l}^{5}-1352\,{l}^{4}+504\,{l}^{3}-32\,{l}^{2})x_{3}^{6}\\ -4(2k+l-2)((128\,l-108){k}^{5}+(582\,{l}^{2}-725\,l+144){k}^{4}+(921\,{l}^{3}-1857\,{l}^{2}+800\,l-48){k}^{3}+(736\,{l}^{4}-2072\,{l}^{3}+1600\,{l}^{2}-324\,l){k}^{2}+(312\,{l}^{5}-1152\,{l}^{4}+1292\,{l}^{3}-484\,{l}^{2}+48\,l)k+56\,{l}^{6}-268\,{l}^{5}+424\,{l}^{4}-252\,{l}^{3}+40\,{l}^{2})x_{3}^{5}\\ +((200\,l-320){k}^{6}+(2448\,{l}^{2}-4584\,l+1712){k}^{5}+(6812\,{l}^{3}-18568\,{l}^{2}+13190\,l-2216){k}^{4}+(8618\,{l}^{4}-32023\,{l}^{3}+36747\,{l}^{2}-13648\,l+1056){k}^{3}+(5781\,{l}^{5}-27998\,{l}^{4}+45785\,{l}^{3}-29060\,{l}^{2}+6104\,l-160){k}^{2}+(2000\,{l}^{6}-12260\,{l}^{5}+27136\,{l}^{4}-25936\,{l}^{3}+9932\,{l}^{2}-1120\,l)k+280\,{l}^{7}-2120\,{l}^{6}+6124\,{l}^{5}-8336\,{l}^{4}+5284\,{l}^{3}-1280\,{l}^{2}+64\,l)x_{3}^{4}\\ -(2(l-2))(2k+l-2)((128\,l-108){k}^{4}+(582\,{l}^{2}-725\,l+144){k}^{3}+(842\,{l}^{3}-1694\,{l}^{2}+716\,l-48){k}^{2}+(512\,{l}^{4}-1560\,{l}^{3}+1232\,{l}^{2}-244\,l)k+112\,{l}^{5}-480\,{l}^{4}+632\,{l}^{3}-280\,{l}^{2}+32\,l)x_{3}^{3}\\ +2(l-2)^{2}(20\,{k}^{5}+(241\,l-125){k}^{4}+(659\,{l}^{2}-800\,l+158){k}^{3}+(703\,{l}^{3}-1593\,{l}^{2}+820\,l-68){k}^{2}+(328\,{l}^{4}-1156\,{l}^{3}+1168\,{l}^{2}-336\,l+8)k+56\,{l}^{5}-276\,{l}^{4}+448\,{l}^{3}-268\,{l}^{2}+48\,l)x_{3}^{2}\\ -2\,(l-2)^{3}(2\,k+l-2)(-2+3\,k+2\,l)(3\,{k}^{2}+2k(6l-1)+4l(2l-1))x_{3}\\ +(l-2)^{4}(k+l)(-2+3\,k+2\,l)^{2}.$

It is not hard to verify that the coefficients of polynomial $p(x_{3})$ is positive for even degree and negative for odd degree whenever $l>k\geq 3$. That is, all the solution of $x_{3}$ for homogeneous Einstein equations are positive (if exist) whenever $l>k\geq 3$.

In summary, for $l>k\geq 3$, we get two solutions of the form

[TABLE]

where $\alpha(x_{12})$ is a rational polynomial of $x_{12}$ with positive values. By Proposition 2.4, every Einstein metric induced by the solution of this form is non-naturally reductive. Thus, we have proved Theorem 3.1.

By Theorem 3.1, for any $3\leq k\leq[\frac{n-1}{3}]$, ${\mathrm{S}}{\mathrm{O}}(n)$ admits at least two non-naturally reductive left-invariant Einstein metrics which are ${\mathrm{A}}{\mathrm{d}}({\mathrm{S}}{\mathrm{O}}(k)\times{\mathrm{S}}{\mathrm{O}}(k)\times{\mathrm{S}}{\mathrm{O}}(n-2k))$-invariant. That is, Theorem 1.1 follows.