Editing Cartan subalgebra (section)

== Existence and uniqueness ==
Cartan subalgebras exist for finite-dimensional Lie algebras whenever the base [[field (mathematics)|field]] is infinite. One way to construct a Cartan subalgebra is by means of a [[regular element of a Lie algebra#A Cartan subalgebra and a regular element|regular element]]. Over a finite field, the question of the existence is still open.{{citation needed|date=January 2020}}

For a finite-dimensional semisimple Lie algebra <math>\mathfrak g</math> over an algebraically closed field of characteristic zero, there is a simpler approach: by definition, a [[toral subalgebra]] is a subalgebra of <math>\mathfrak g</math> that consists of semisimple elements (an element is semisimple if the [[adjoint endomorphism]] induced by it is [[diagonalizable matrix|diagonalizable]]). A Cartan subalgebra of <math>\mathfrak g</math> 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 [[automorphism]]s of the algebra, and in particular are all [[isomorphism|isomorphic]]. The common dimension of a Cartan subalgebra is then called the [[rank of a Lie algebra|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.<ref>{{harvnb|Hall|2015}} Chapter 7</ref> In that case, <math>\mathfrak{h}</math> may be taken as the complexification of the Lie algebra of a [[maximal torus]] of the compact group.

If <math>\mathfrak{g}</math> is a [[linear Lie algebra]] (a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space ''V'') over an algebraically closed field, then any Cartan subalgebra of <math>\mathfrak{g}</math> is the [[centralizer]] of a maximal [[toral Lie algebra|toral subalgebra]] of <math>\mathfrak{g}</math>.{{citation needed|date=January 2020}} If <math>\mathfrak{g}</math> is semisimple and the field has characteristic zero, then a maximal toral subalgebra is self-normalizing, and so is equal to the associated Cartan subalgebra.  If in addition <math>\mathfrak g</math> is semisimple, then the [[Adjoint representation of a Lie group|adjoint representation]] presents <math>\mathfrak g</math> as a linear Lie algebra, so that  a subalgebra of <math>\mathfrak g</math> is Cartan if and only if it is a maximal toral subalgebra.