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.