Cartan subalgebra

In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic zero, a Cartan subalgebra can also be defined as a maximal toral subalgebra.

(As noted earlier, a Cartan subalgebra can in fact be characterized as a subalgebra that is maximal among those having the above two properties.)

Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.