Maximal torus

Summary

A maximal torus of a compact Lie group is a subgroup isomorphic to a torus $\mathbb{T}^k$ that is not contained in any larger such subgroup. In a compact connected Lie group all maximal tori are conjugate, and every element lies in one of them. Much of the theory of compact Lie groups is read off from a maximal torus and the action of the Weyl group on it.