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).^[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 ^b Humphreys 1972, Ch. II, § 8.1.
^ Proof (from Humphreys): Let $x\in {\mathfrak {h}}$ . Since $\operatorname {ad} (x)$ is diagonalizable, it is enough to show the eigenvalues of $\operatorname {ad} _{\mathfrak {h}}(x)$ are all zero. Let $y\in {\mathfrak {h}}$ be an eigenvector of $\operatorname {ad} _{\mathfrak {h}}(x)$ with eigenvalue $\lambda$ . Then $x$ is a sum of eigenvectors of $\operatorname {ad} _{\mathfrak {h}}(y)$ and then $-\lambda y=\operatorname {ad} _{\mathfrak {h}}(y)x$ is a linear combination of eigenvectors of $\operatorname {ad} _{\mathfrak {h}}(y)$ with nonzero eigenvalues. But, unless $\lambda =0$ , we have that $-\lambda y$ is an eigenvector of $\operatorname {ad} _{\mathfrak {h}}(y)$ with eigenvalue zero, a contradiction. Thus, $\lambda =0$ . $\square$