[proofplan]
We first check that the Weyl [group action](/page/Group%20Action) on $T$ is well-defined and that the proposed assignment from Weyl orbits to conjugacy classes does not depend on the representative of the orbit. Surjectivity follows from the [maximal torus theorem](/theorems/9713), which says that every element of $G$ is conjugate to an element of $T$. Injectivity follows from the theorem that two elements of $T$ which are conjugate in $G$ lie in the same Weyl orbit.
[/proofplan]
[step:Verify that the Weyl group acts on the maximal torus by conjugation]
Let $N_G(T):=\{n\in G:nTn^{-1}=T\}$ denote the normalizer of $T$ in $G$. For $n\in N_G(T)$ and $t\in T$, the element $ntn^{-1}$ lies in $T$ by the defining property of $N_G(T)$, so conjugation by $n$ defines a map from $T$ to $T$.
We check that the formula
\begin{align*}
(nT)\cdot t:=ntn^{-1}
\end{align*}
does not depend on the representative $n$ of the coset $nT$. If $n'\in N_G(T)$ satisfies $n'T=nT$, then there exists $\tau\in T$ such that $n'=n\tau$. Since $T$ is abelian, for every $t\in T$ we have
\begin{align*}
n't(n')^{-1}=n\tau t\tau^{-1}n^{-1}=ntn^{-1}.
\end{align*}
Thus the action of $W=N_G(T)/T$ on $T$ is well-defined.
[/step]
[step:Show that the assignment from Weyl orbits to conjugacy classes is well-defined]
Define
\begin{align*}
\Phi:T/W&\to \{\operatorname{Cl}_G(g):g\in G\}
\end{align*}
\begin{align*}
W\cdot t&\mapsto \operatorname{Cl}_G(t).
\end{align*}
Suppose $s,t\in T$ lie in the same $W$-orbit. Then there exists $n\in N_G(T)$ such that
\begin{align*}
s=ntn^{-1}.
\end{align*}
Since $n\in G$, this equality says precisely that $s$ and $t$ are conjugate in $G$. Hence
\begin{align*}
\operatorname{Cl}_G(s)=\operatorname{Cl}_G(t).
\end{align*}
Therefore $\Phi$ is independent of the representative $t$ chosen from the orbit $W\cdot t$.
[/step]
[step:Use the maximal torus theorem to prove surjectivity]
Let $\operatorname{Cl}_G(g)$ be an arbitrary [conjugacy class](/page/Conjugacy%20Class) in $G$, with $g\in G$. Since $G$ is compact and connected and $T$ is a maximal torus, the [Maximal Torus Theorem][citetheorem:9713] gives elements $h\in G$ and $t\in T$ such that
\begin{align*}
hgh^{-1}=t.
\end{align*}
Thus $g$ and $t$ are conjugate in $G$, so
\begin{align*}
\operatorname{Cl}_G(g)=\operatorname{Cl}_G(t)=\Phi(W\cdot t).
\end{align*}
Since the conjugacy class $\operatorname{Cl}_G(g)$ was arbitrary, $\Phi$ is surjective.
[/step]
[step:Use the Weyl orbit criterion inside $T$ to prove injectivity]
Suppose $s,t\in T$ satisfy
\begin{align*}
\Phi(W\cdot s)=\Phi(W\cdot t).
\end{align*}
By definition of $\Phi$, this means
\begin{align*}
\operatorname{Cl}_G(s)=\operatorname{Cl}_G(t).
\end{align*}
Hence $s$ and $t$ are conjugate in $G$. Since $s,t\in T$, the theorem that [conjugacy classes meet a maximal torus in one Weyl orbit](/theorems/9724), namely [Conjugacy Classes Meet a Maximal Torus in One Weyl Orbit][citetheorem:9724], implies that $s$ and $t$ lie in the same $W$-orbit in $T$. Therefore
\begin{align*}
W\cdot s=W\cdot t.
\end{align*}
Thus $\Phi$ is injective.
[guided]
We want to prove injectivity of $\Phi$, so we start with two Weyl orbits that have the same image under $\Phi$. Let $s,t\in T$ and assume
\begin{align*}
\Phi(W\cdot s)=\Phi(W\cdot t).
\end{align*}
The definition of $\Phi$ turns this equality of images into an equality of conjugacy classes:
\begin{align*}
\operatorname{Cl}_G(s)=\operatorname{Cl}_G(t).
\end{align*}
By the definition of conjugacy class, this means that $s$ and $t$ are conjugate in $G$: there exists an element $g\in G$ such that
\begin{align*}
s=gtg^{-1}.
\end{align*}
At this point the important issue is that $g$ need not lie in $N_G(T)$. If all we knew were conjugacy by an arbitrary element of $G$, we would not yet know that $s$ and $t$ are in the same Weyl orbit, because Weyl orbits are defined using representatives from $N_G(T)$. The needed compact Lie group input is exactly [Conjugacy Classes Meet a Maximal Torus in One Weyl Orbit][citetheorem:9724]. Its hypotheses apply here: $G$ is compact and connected, $T$ is a maximal torus, and $s,t$ are elements of $T$ which are conjugate in $G$. Therefore the theorem gives an element $n\in N_G(T)$ such that
\begin{align*}
s=ntn^{-1}.
\end{align*}
This equality is precisely the statement that $s$ lies in the $W$-orbit of $t$ under the action of $W=N_G(T)/T$ on $T$. Hence
\begin{align*}
W\cdot s=W\cdot t.
\end{align*}
So equal images under $\Phi$ force equal Weyl orbits, which proves that $\Phi$ is injective.
[/guided]
[/step]
[step:Conclude that the orbit assignment is a bijection]
The map $\Phi:T/W\to \{\operatorname{Cl}_G(g):g\in G\}$ is well-defined by the preceding verification. It is surjective because every conjugacy class in $G$ has a representative in $T$, and it is injective because two representatives in $T$ determine the same conjugacy class exactly when they lie in the same Weyl orbit. Therefore $\Phi$ is a bijection.
[/step]
