Maximal Torus

A $k$-dimensional torus is ${\mathbb{T}}^k={\mathbb{S}}^1\times {\mathbb{S}}^1\times \cdots \times {\mathbb{S}}^1$ (k-fold Cartesian product) (note that any torus is abelian)

Definition

The maximal torus of a lie group $G$ is the maximal subgroup (not necessarily unique) isomorphic to ${\mathbb{T}}^k$

Example

• The maximal torus of $SO(2)$ is itself, because $SO(2)\cong {\mathbb{S}}^1\cong {\mathbb{T}}^1$
• The maximal torus of $SO(3)$ is also ${\mathbb{T}}^1$