Corollary 8

Let H be a compact Lie group, T a maximal torus of H. Then $H = N_H(T).H_0$, and the injection of $N_H(T)$ into H induces an isomorphism from $N_H(T)/N_{H_0}(T)$ to $H/H_0$.

Let $h\in H$. Then $h^{-1}Th$ is a maximal torus of $H_0$, hence (Th. 2) there exists $g\in H_0$ such that $hg\in N_H(T)$; thus $h$ belongs to $N_H(T).H_0$, hence the first assertion. The second follows immediately.