Cartan subgroup

Notice that in the context of algebraic groups a torus is an algebraic group ${\displaystyle T}$ such that the base extension ${\displaystyle T_{(\overline{k})}}$ (where ${\displaystyle \overline{k}}$ is the algebraic closure of ${\displaystyle k}$) is isomorphic to the product of a finite number of copies of the ${\displaystyle {\mathbf{G}}_m={\mathbf{G}\mathbf{L}}_1}$. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal tori in the theory of Lie groups.

If ${\displaystyle G}$ is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser and thus Cartan subgroups of ${\displaystyle G}$ are precisely the maximal tori.

Example

The general linear groups ${\displaystyle {\mathbf{G}\mathbf{L}}_n}$ are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of ${\displaystyle {\mathbf{G}}_m}$ already before any base extension), and it can be shown to be maximal. Since ${\displaystyle {\mathbf{G}\mathbf{L}}_n}$ is reductive, the diagonal subgroup is a Cartan subgroup.