In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algebra is abelian; thus, its elements are simultaneously diagonalizable.

In semisimple and reductive Lie algebras

A subalgebra \mathfrak h of a semisimple Lie algebra \mathfrak g is called toral if the adjoint representation of \mathfrak h on \mathfrak g, is a toral subalgebra. A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra, over an algebraically closed 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 \mathfrak g restricted to \mathfrak h is nondegenerate. For more general Lie algebras, a Cartan subalgebra may differ from a maximal toral subalgebra. In a finite-dimensional semisimple Lie algebra \mathfrak g over an algebraically closed field of a characteristic zero, a toral subalgebra exists. In fact, if \mathfrak g has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence, \mathfrak g must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.