In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic . In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the adjoint endomorphism is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.pg 231 In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic zero, a Cartan subalgebra can also be defined as a maximal toral subalgebra. Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero). Existence and uniqueness Cartan subalgebras exist for finite-dimensional Lie algebras whenever the base field is infinite. One way to construct a Cartan subalgebra is by means of a regular element. Over a finite field, the question of the existence is still open. For a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero, there is a simpler approach: by definition, a toral subalgebra is a subalgebra of that consists of semisimple elements (an element is semisimple if the adjoint endomorphism induced by it is diagonalizable). A Cartan subalgebra of is then the same thing as a maximal toral subalgebra and the existence of a maximal toral subalgebra is easy to see. In a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero, all Cartan subalgebras are conjugate under automorphisms of the algebra, and in particular are all isomorphic. The common dimension of a Cartan subalgebra is then called the rank of the algebra. For a finite-dimensional complex semisimple Lie algebra, the existence of a Cartan subalgebra is much simpler to establish, assuming the existence of a compact real form. In that case, may be taken as the complexification of the Lie algebra of a maximal torus of the compact group. If is a linear Lie algebra (a Lie subalgebra of the Lie algebra of endomorphisms of