Cartan subalgebra

Summary

A Cartan subalgebra of a Lie algebra $\mathfrak g$ is a nilpotent subalgebra that is equal to its own normalizer. For a complex semisimple Lie algebra, the Cartan subalgebras are the maximal abelian subalgebras made of semisimple elements, and all of them are conjugate. A Cartan subalgebra $\mathfrak h$ acts diagonally on $\mathfrak g$, and the eigenvalues of this action, the roots, form a root system. This is the starting point of the classification of semisimple Lie algebras and of highest weight theory.