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 .

Theorem (maximal tori)

1. (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.

2. (Conjugacy) Any two maximal tori $T_1,T_2\subseteq G$ are conjugate: there exists $g\in G$ such that

   $$T_2=gT_1g^{-1}.$$

3. (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.