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).
[1] Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements.
Over an algebraically closed field, every toral Lie algebra is abelian;[1][2] thus, its elements are simultaneously diagonalizable.
A maximal toral Lie subalgebra of a finite-dimensional semisimple Lie algebra, or more generally of a finite-dimensional reductive Lie algebra,[citation needed] over an algebraically closed field of characteristic 0 is a Cartan subalgebra and vice versa.
[3] In particular, a maximal toral Lie subalgebra in this setting is self-normalizing, coincides with its centralizer, and the Killing form of
In a finite-dimensional semisimple Lie algebra
over an algebraically closed field of a characteristic zero, a toral subalgebra exists.
has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity.
must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.