Let $G$ be a connected Lie group and let $K\subseteq G$ be a maximal compact subgroup. If $H\subseteq G$ is any compact subgroup I want to show that there exists $g\in G$ such that $g^{-1}Hg\subseteq K$.
Proof Idea: Note that $H$ acts on the coset space $G/K$ by left multiplication. We will be done if we can find a global fixed point $gK$ such that $hgK=gK$ (and hence $g^{-1}hg\in K$) for all $h\in H$.
My Question: What general properties of this action guarantee that it has a fixed point? I see hints that this might be related to the Bruhat-Tits fixed point theorem.