= Adjoint-orbit criterion for maximal-torus conjugacy
Give the <Lie algebra> of a <compact Lie group> an <inner product> invariant under the <Adjoint representation>. For a <maximal torus> $T$ with Lie algebra $\mathfrak t$, its Lie-algebra centralizer is $\mathfrak t$: any additional commuting vector would generate, together with $T$, a larger compact connected abelian subgroup. The torus weight decomposition therefore supplies $H\in\mathfrak t$ whose centralizer is exactly $\mathfrak t$, by avoiding finitely many nonzero-weight hyperplanes. For any $X$, maximize $\langle\operatorname{Ad}_gX,H\rangle$ on the group. Its directional derivative gives $\langle Z,[\operatorname{Ad}_gX,H]\rangle=0$ for every $Z$, so $\operatorname{Ad}_gX\in\mathfrak t$. For a connected compact group, the <bi-invariant Riemannian metric> and <Hopf-Rinow theorem> make the exponential onto; exponentiating this conjugacy puts every element in a conjugate of $T$.
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