I am sorry if this question is too elementary to be posted here, but no experts answer this question when I post it on Math Stackexchange.


Let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be a Cartan decomposition for a noncompact real simple Lie algebra $\mathfrak{g}$ corresponding to a Cartan involution $\theta$, where $\mathfrak{k}$ is the maximal compact subalgebra of $\mathfrak{g}$. Suppose that $\sigma$ is another involutive automorphism of $\mathfrak{g}$ such that $\sigma\theta=\theta\sigma$. Then $\sigma$ preserves the Cartan decomposition, and $\sigma|\_\mathfrak{k}:\mathfrak{k}\rightarrow\mathfrak{k}$. By the classification for symmetric pairs, it seems true that $\sigma|\_\mathfrak{k}$ is never the identity map, but how to prove this fact theoretically (instead of case by case)?


In other words, there is no (non-Cartan) involutive automorphism $\sigma$ of a noncompact real simple Lie algebra $\mathfrak{g}$ such that the subalgebra $\mathfrak{g}^\sigma$ of the fixed points under the action of $\sigma$ on $\mathfrak{g}$ contains a maximal compact subalgebra. How to prove it? I shall be grateful if experts here may offer any hint.



Involutive automorphism of simple Lie algebra
Let $\mathfrak g$ be a noncompact simple Lie algebra and let $\mathfrak g=\mathfrak k+\mathfrak p$ be a Cartan decomposition. The simplicity of $\mathfrak g$ implies that
the adjoint representation of $\mathfrak k$ on $\mathfrak p$ is irreducible (indeed,
if $\mathfrak p\_1$ is an $\mathrm{ad}\_\mathfrak k$-invariant subspace of
$\mathfrak p$, one can show$^\dagger$ that $\mathfrak g\_1:=[\mathfrak p\_1,\mathfrak p\_1]+\mathfrak p\_1$ is an ideal of $\mathfrak g$). In particular, $\mathfrak k$ is a *maximal* (not only maximal compact) subalgebra of $\mathfrak g$. Indeed, if $\mathfrak h$ were a subalgebra containing $\mathfrak k$, then
$\mathfrak h =\mathfrak k +\mathfrak h\cap\mathfrak p$ and $\mathfrak h\cap\mathfrak p$ would be an $\mathrm{ad}\_{\mathfrak k}$-invariant subspace of $\mathfrak p$.


$\dagger$ Write $\mathfrak p=\mathfrak p\_1+\mathfrak p\_2$ $\mathrm{ad}\_{\mathfrak k}$-invariant decomposition. Let $X\_i\in\mathfrak p\_i$ and $Y=[X\_1,X\_2]\in\mathfrak k$. Denote the Cartan-Killing form of $\mathfrak g$ by $B$; it is negative definite on $\mathfrak k$ and positive definite on $\mathfrak p$; in particular, we may assume $B(\mathfrak p\_1,\mathfrak p\_2)=0$. Now $B(Y,Y)=B(Y,[X\_1,X\_2])=B([Y,X\_1],X\_2)\in B(\mathfrak p\_1,\mathfrak p\_2)=0$, so $Y=0$. We have shown that $[\mathfrak p\_1,\mathfrak p\_2]=0$. Using Jacobi, one completes the check that $\mathfrak g\_1$ is an ideal of $\mathfrak g$.


