In mathematics, a toral Lie algebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are diagonalizable (or semisimple). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Every toral Lie algebra is abelian; thus, its members are simultaneously diagonalizable.
A subalgebra H of a semisimple Lie algebra L is called toral if the adjoint representation of H on L, ad(H)⊂gl(L) is a toral Lie algebra. A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra, over a field of characteristic 0 is a Cartan subalgebra and vice versa. In particular, a maximal toral Lie subalgebra in this setting is self-normalizing, coincides with its centralizer, and the Killing form of L restricted to H is nondegenerate.
For more general Lie algebras, a Cartan algebra may differ from a maximal toral algebra.