Let $G$ be a compact connected Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$. Let $T\le G$ be a maximal torus, and let $\mathfrak t\le \mathfrak g$ be the Lie algebra of $T$. Then $\mathfrak t$ is a maximal abelian Lie subalgebra of $\mathfrak g$: if $\mathfrak a\le \mathfrak g$ is an abelian Lie subalgebra with $\mathfrak t\subset \mathfrak a$, then $\mathfrak a=\mathfrak t$.