Cartan subgroup

In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group $$G$$ over a (not necessarily algebraically closed) field $$k$$ is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If $$k$$ is algebraically closed, they are all conjugate to each other.

Notice that in the context of algebraic groups a torus is an algebraic group $$T$$ such that the base extension $$T_{(\bar{k})}$$ (where $$\bar{k}$$ is the algebraic closure of $$k$$) is isomorphic to the product of a finite number of copies of the $$\mathbf{G}_m=\mathbf{GL}_1$$. 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 $$G$$ 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 $$G$$ are precisely the maximal tori.

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