Corollary 1

Assume that the $\mathbf{Z}$-module $\pi_1(G)$ is torsion-free.

a) The centralizer of every element of G is connected.

b) Any two commuting elements of G belong to the same maximal torus.

By Cor. 3 of Prop. 11 of §4, no. 6, D(G) is simply-connected. We have $G = C(G)_0.D(G)$; let $x\in G$; write $x=uv$, with $u\in C(G)_0$ and $v\in D(G)$. Then $Z(x) = C(G)_0.Z_{D(G)}(v)$. By Th. $1c), Z_{D(G)}(v)$ is connected, so $Z(x)$ is connected, hence a). Thus, by Cor. 3 of Th. 2 of §2, no. $2, Z(x)$ is the union of the maximal tori of G containing $x$, hence b).