Conjugacy of maximal tori

Remarks

One proof uses the Lefschetz fixed point theorem on the manifold $G/T$. For $x \in G$, the map $gT \mapsto xgT$ of $G/T$ is homotopic to the identity, since $G$ is connected. So its Lefschetz number is the Euler characteristic of $G/T$, which equals the order of the Weyl group and is nonzero. A fixed point $gT$ means $xgT = gT$, that is, $g^{-1}xg \in T$; this gives part 2, and part 1 follows by taking for $x$ a generator of a dense cyclic subgroup of $T'$. Example: for $G = U(n)$ and $T$ the diagonal unitary matrices, part 2 is the spectral theorem: every unitary matrix is $g D g^{-1}$ with $g$ unitary and $D$ diagonal. Connectedness is needed for part 2: in $O(2)$, every torus lies in the identity component $SO(2)$, so a reflection lies in no maximal torus.

Going deeper

For a student

Part 2 implies that a continuous class function on $G$ is determined by its restriction to $T$, and that restriction is invariant under the Weyl group. This is why characters of representations are studied on $T$, and why the Weyl integration formula reduces integrals of class functions to integrals over $T$. It also implies that the exponential map of a compact connected Lie group is surjective, since it is surjective on each torus.