In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group
over a (not necessarily algebraically closed) field
Cartan subgroups are smooth (equivalently reduced), connected and nilpotent.
is algebraically closed, they are all conjugate to each other.
[1] Notice that in the context of algebraic groups a torus is an algebraic group
) 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.
is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser [2] and thus Cartan subgroups of
already before any base extension), and it can be shown to be maximal.
This algebraic geometry–related article is a stub.