Proposition 5

$a)$ Every conjugacy class of G meets T.

b) The intersections with T of the conjugacy classes of G are the orbits of the Weyl group.

Let $g\in G$; by Th. 2 of no. 2, there exists $h\in G$ such that $g\in hTh^{-1}$, hence a). By definition of the Weyl group, any two elements in the same orbit of $W_G(T)$ on T are conjugate in G; conversely, let $a, b$ be two elements of T conjugate under G. There exists $h\in G$ such that $b=hah^{-1}$; applying Cor. 7 of Th. 2 (no. 2) with $A =\{a\},s=$ Int $h, T'= T$, we see that there exists $g\in G$ such that Int $hg$ maps T to T and $a$ to $b$. The class of $hg$ in $W_G(T)$ then maps $a$ to $b$, hence the proposition.