On stable Cartan subgroups of Lie groups

Let $G$ be a connected real Lie group with associated Lie algebra $\mathfrak g$, and let ${\rm Aut}(G)$ be the group of (Lie) automorphisms of $G$. It is noted here that, given a super-solvable subgroup $\Gamma\subset {\rm Aut}(G)$ of semisimple automorphisms, there exists a $\Gamma$-stable Cartan subgroup, by using a result of Borel and Mostow. We characterize the $\Gamma$-stable Cartan subgroups (with induced action) in the quotient group modulo a $\Gamma$-stable closed normal subgroup as the images of the $\Gamma$-stable Cartan subgroups in the ambient group. It is well known that a semisimple automorphism of $\mathfrak g$ always fixes a Cartan subalgebra of $\mathfrak g$. Conversely, if we take a representative from each non-conjugate class of Cartan subalgebras in a real Lie algebra, we show that there exists a non-identity automorphism that fixes these representatives. We explicitly identify such automorphisms in the case of classical simple Lie algebras. As a consequence, we deduce an analogous result for semisimple Lie groups. Moreover, given a $\Gamma$-stable Cartan subgroup $H$ of $G$, and a $\Gamma$-stable closed connected normal subgroup $M$ of $G$, we prove that there exists a $\Gamma$-stable Cartan subgroup $H_M$ of $M$ such that $H\cap M\subset H_M$.