Corollary

Assume that G is semi-simple. Then every element of G is the commutator of two elements of G.

Let $c$ be a Coxeter transformation of the Weyl group $W(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$ (Chap. V, §6, no. 1), and let $n$ be an element of $N_G(T)$ whose class in W is identified with $c$ by the isomorphism defined in the proposition. Denote by $f_c$ the morphism $t \rightarrow (n, t)$ from T to T; for $x\in \mathfrak{t}_{\mathbf{C}}$, we have $L(f_c)_{(\mathbf{C})}(x) =$ (Ad $n$)$(x)-x=$ $c(x)-x$.

By Th. 1 of Chap. V, §6, no. 2, the endomorphism $c$ of $\mathfrak{t}_{\mathbf{C}}$ has no eigenvalue equal to 1. Consequently, $L(f_c)$ is surjective, and hence so is $f_c$. It follows that every element of T is the commutator of two elements of G, which implies the corollary in view of Th. 2, §2, no. 2.