Let
be a finite-dimensional Lie algebra over some field
. A subalgebra of is called a Cartan subalgebra if it is nilpotent
and equal to its normalizer, which is the set of those elements such that .

It follows from the definition that if is nilpotent, then itself is a Cartan subalgebra of . On the other hand, let be the Lie algebra of all endomorphisms
of
(for some natural number ), with . Then the set of all endomorphisms of of the form is a Cartan subalgebra
of .

It can be proved that:

1. If
is infinite, then
has Cartan subalgebras.

2. If the characteristic of is equal to , then all Cartan subalgebras of have the same dimension.

4. If
is semisimple and
is an infinite field whose characteristic
is equal to 0, then all Cartan subalgebras of are Abelian.

Every Cartan subalgebra of a Lie algebra is a maximal nilpotent subalgebra of . However, a maximal nilpotent subalgebra of doesn't have to be a Cartan subalgebra. For instance, if is the Lie algebra of all endomorphisms
of
with
and if
is the subalgebra of all endomorphisms of the form , then is a maximal nilpotent subalgebra of , but not a Cartan subalgebra.