Let $G$ be a compact connected Lie group with identity element $e$, and let $T\le G$ be a maximal torus, meaning a compact connected abelian Lie subgroup maximal among compact connected abelian Lie subgroups of $G$. Then for every $g\in G$ there exist $h\in G$ and $t\in T$ such that $hgh^{-1}=t$. Equivalently, every element of $G$ is conjugate in $G$ to an element of $T$.