Maximal torus theorem

Let $G$ be a compact Lie group that is also connected. A torus means a Lie group isomorphic to $(S^1)^r$ ; see the torus example and the structure of connected abelian Lie groups.

(Existence) $G$ contains a maximal torus $T\subseteq G$ , i.e. a connected, compact, abelian Lie subgroup not properly contained in any larger connected, compact, abelian subgroup.

(Conjugacy) Any two maximal tori $T_1,T_2\subseteq G$ are conjugate: there exists $g\in G$ such that $T_2=gT_1g^{-1}.$

(Every element lies in a maximal torus) For every $x\in G$ there exists a maximal torus $T\subseteq G$ with $x\in T$ .

Why it matters

A maximal torus is the compact-group analogue of a Cartan subalgebra: its Lie algebra $\mathfrak t=\operatorname{Lie}(T)$ is a maximal abelian subalgebra of $\mathfrak g=\operatorname{Lie}(G)$ consisting of semisimple elements (over $\Bbb C$ ). The conjugacy statement implies that many structural invariants of $G$ can be computed from $\mathfrak t$ up to the action of the Weyl group, leading to root data and classification results.