Conjugacy classes meet T in Weyl orbits

Let $x\in G$. By [L1] there is a maximal torus $M$ with $x\in M$, and by [L2] there is $g\in G$ with $M=gT{g}^{-1}$; then $g^{-1}xg\in T$, so the conjugacy class of $x$ meets $T$.

Suppose $t^{\mathrm{\prime }}=gt{g}^{-1}$ for some $g\in G$. Put $M:=gT{g}^{-1}$, a maximal torus containing $t^{\mathrm{\prime }}$. Both $T$ and $M$ lie in $C_G(t^{\mathrm{\prime }})$ and, being connected and containing the identity, lie in the compact connected Lie group $H:=C_G(t^{\mathrm{\prime }}{)}^0$ by [L4]. A torus of $H$ properly containing $T$ or $M$ would also be a torus of $G$ properly containing a maximal torus of $G$; hence $T$ and $M$ are maximal tori of $H$.

By [L2] applied to the compact connected Lie group $H$, there is $h\in H$ with $hT{h}^{-1}=M=gT{g}^{-1}$. Then $n:=g^{-1}h\in N_G(T)$, and since $h$ centralizes $t^{\mathrm{\prime }}$ we obtain $t^{\mathrm{\prime }}=h^{-1}{t}^{\mathrm{\prime }}h=h^{-1}(gt{g}^{-1})h=n^{-1}tn$; hence $t^{\mathrm{\prime }}$ lies in the $W(G,T)$-orbit of $t$ by [L3].

Conversely, if $t^{\mathrm{\prime }}=nt{n}^{-1}$ for some $n\in N_G(T)$, then $t^{\mathrm{\prime }}=nt{n}^{-1}$ is conjugate to $t$ and lies in $T$; hence conjugacy in $G$ between points of $T$ is exactly the orbit relation of the Weyl group action, and by step 1.1 every conjugacy class meets $T$. The Axiom of Choice entered through [L1], [L2], and the countable-choice closed-subgroup input in [L4].