Toral subalgebra
In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple. 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 of a semisimple Lie algebra is called toral if the adjoint representation of on, 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 restricted to is nondegenerate.
For more general Lie algebras, a Cartan algebra may differ from a maximal toral algebra.
In a finite-dimensional semisimple Lie algebra over an algebraically closed field of a characteristic zero, a toral subalgebra exists. In fact, if has only nilpotent elements, then it is nilpotent, but then its Killing form is identically zero, contradicting semisimplicity. Hence, must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.
