Let $\fr g$ be a Lie algebra. Then a Cartan subalgebra is a maximal subalgebra of $\fr g$ which is self-normalizing, that is, if $[g,h]\in\fr h$ for all $h\in\fr h$ , then $g\in\fr h$ as well. Any Cartan subalgebra $\fr h$ is nilpotent, and if $\fr g$ is semi-simple, it is abelian. All Cartan subalgebras of a Lie algebra are conjugate by the adjoint action of any Lie group with algebra $\fr g$ .

The dimension of $\fr h$ is called the rank of $\fr g$ .