Corollary 5

a) Let S be a torus of G. The centralizer of S is connected; it is the union of the maximal tori of G containing S.

b) Let $\mathfrak{s}$ be a commutative subalgebra of L(G). The fixer of $\mathfrak{s}$ in G is connected; it is the union of the maximal tori of G whose Lie algebras contain $\mathfrak{s}$.

To prove a), it suffices to prove that if an element $g$ of G centralizes S, there exists a maximal torus of G containing S and $g$. Now, if C is the centralizer of $g$, we have $g\in C_0$ (Cor. 3) and $S\subset C_0$; if T is a maximal torus of the connected compact Lie group $C_0$ containing S, we have $g\in T$ (Cor. 2), hence a). Assertion b) follows from a) applied to the closure of the integral subgroup with Lie algebra $\mathfrak{s}$, in view of Chap. III, §9, no. 3, Prop. 9.