Toral subalgebra

In semisimple and reductive Lie algebras

A subalgebra ${\displaystyle \mathfrak{h}}$ of a semisimple Lie algebra ${\displaystyle \mathfrak{g}}$ is called toral if the adjoint representation of ${\displaystyle \mathfrak{h}}$ on ${\displaystyle \mathfrak{g}}$, ${\displaystyle \mathrm{ad}(\mathfrak{h})\subset \mathfrak{g}\mathfrak{l}(\mathfrak{g})}$ 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 ${\displaystyle \mathfrak{g}}$ restricted to ${\displaystyle \mathfrak{h}}$ is nondegenerate.

For more general Lie algebras, a Cartan subalgebra may differ from a maximal toral subalgebra.

In a finite-dimensional semisimple Lie algebra ${\displaystyle \mathfrak{g}}$ over an algebraically closed field of a characteristic zero, a toral subalgebra exists. In fact, if ${\displaystyle \mathfrak{g}}$ has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence, ${\displaystyle \mathfrak{g}}$ must have a nonzero semisimple element, say x; the linear span of x is then a toral subalgebra.