Cartan subalgebra

Let $\mathfrak{g}$ be a Lie algebra. Then a Cartan subalgebra is a maximal subalgebra of $\mathfrak{g}$ which is self-normalizing, that is, if $[g,h]\in\mathfrak{h}$ for all $h\in\mathfrak{h}$ , then $g\in\mathfrak{h}$ as well. Any Cartan subalgebra $\mathfrak{h}$ is nilpotent, and if $\mathfrak{g}$ 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 $\mathfrak{g}$ .

The dimension of $\mathfrak{h}$ is called the rank of $\mathfrak{g}$ .