Toral subalgebra

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 $$\mathfrak g$$, $$\operatorname{ad}(\mathfrak h) \subset \mathfrak{gl}(\mathfrak g)$$ 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.