Cartan subalgebra

Definition
Given a Lie algebra over a field, a Cartan subalgebra is a nilpotent Lie subalgebra that equals its own normalizer in the whole Lie algebra.

In a semisimple Lie algebra, the Cartan subalgebras are precisely the maximal abelian Lie subalgebras.

Facts
Pick a Cartan subalgebra of a semisimple Lie algebra. Then, any representation of the whole Lie algebra can be expressed as a direct sum of subspaces each of which is an eigenspace for every vector in the Cartan subalgebra. Within each subspace, the map taking the Cartan subalgebra to the corresponding eigenvalue is a linear map.

In this context an eigenvalue for the Cartan subalgebra is a linear functional such that there exists an eigenvector for every element of the Cartan subalgebra for which the map sending the Cartan subalgebra to the eigenvalue is the given linear functional. An eigenvalue of the Cartan subalgebra is also termed a weight of the representation.