Proposition 6

a) Every orbit of $\mathrm{G}$ on $\mathfrak{g}$ (for the adjoint representation) meets $\mathfrak{t}$.

b) The intersections with $\mathfrak{t}$ of the orbits of $\mathrm{G}$ are the orbits of $\mathrm{W}_{G}(\mathrm{T})$ on $\mathfrak{t}$.

Assertion a) follows from Th. 1 (no. 1). Let $x,y$ be two elements of $\mathfrak{t}$ conjugate under $\mathrm{Ad}(\mathrm{G})$, and let $h\in\mathrm{G}$ be such that $(\mathrm{Ad}\,h)(x)=y$. Applying the corollary of Th. 1 (no. 1) with $\mathfrak{a}=\{x\}$, $u=\mathrm{Ad}\,h$, $\mathfrak{t}'=\mathfrak{t}$, we see that there exists $g\in\mathrm{G}$ such that $\mathrm{Ad}\,hg$ maps $\mathfrak{t}$ to $\mathfrak{t}$ and $x$ to $y$. Then $hg\in\mathrm{N}_{G}(\mathrm{T})$ (Chap. III, §9, no. 4, Prop. 11), and the class of $hg$ in $\mathrm{W}_{G}(\mathrm{T})$ maps $x$ to $y$, hence the proposition.