Proposition 39

Suppose that $K$ is of characteristic 0. Let $G$ be a Lie group, $E$ a finite-dimensional vector space and $\pi$ an analytic linear representation of $G$ on $E$. Let $E_1, E_2$ be vector subspaces of $E$ such that $E_2 \subset E_1$. The set $G_1$ of $g \in G$ such that $\pi(g)x \equiv x$ (mod. $E_2$) for all $x \in E_1$ is a Lie subgroup of $G$ and $L(G_1)$ is the set of $a \in L(G)$ such that $L(\pi)a$ maps $E_1$ into $E_2$.

This follows from Propositions 29 (no. 8) and 36 (no. 10).