Exercise 13, § 9

Let G be a Lie group, H and K two compact subgroups of G. Assume that G admits a faithful finite dimensional linear representation. Show that there exists a finite set F of subgroups of H such that for all $g\in G$ the subgroup $H\cap gKg^{-1}$ is conjugate in H to a subgroup in F (use Exerc. 5 and 9).