Cartan subgroup

In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group over a field is the centralizer of a maximal torus. Cartan subgroups are smooth, connected and nilpotent. If is algebraically closed, they are all conjugate to each other.
Notice that in the context of algebraic groups a torus is an algebraic group
such that the base extension is isomorphic to the product of a finite number of copies of the. 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 is reductive, then a torus is maximal if and only if it is its own centraliser and thus Cartan subgroups of are precisely the maximal tori.

Example

The general linear groups are reductive. The diagonal subgroup is clearly a torus, and it can be shown to be maximal. Since is reductive, the diagonal subgroup is a Cartan subgroup.