Toral and maximal toral subalgebras

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra. A Lie subalgebra $\mathfrak{t}\subseteq \mathfrak{g}$ (Lie subalgebras, ideals, and center) is toral if it is abelian and ${\mathrm{ad}}_x$ is a semisimple endomorphism of $\mathfrak{g}$ for every $x\in \mathfrak{t}$, semisimplicity being understood in the sense of Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms (Derivations of Lie algebras supplies the notation ${\mathrm{ad}}_x(y)=[x,y]$). Because $\mathfrak{t}$ is abelian, ${\mathrm{ad}}_x{\mathrm{\mid }}_{\mathfrak{t}}$ is the zero endomorphism of $\mathfrak{t}$ and is therefore semisimple automatically: the requirement must be placed on $\mathfrak{g}$ itself, and requiring only that ${\mathrm{ad}}_x{\mathrm{\mid }}_{\mathfrak{t}}$ be semisimple would merely repeat abelianness.

The coordinates on $\mathfrak{g}$ are irrelevant to this notion: the choice of an algebraic closure in Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms makes semisimplicity of an endomorphism a basis-free property, and restricting a semisimple endomorphism to an invariant subspace is again semisimple. A toral subalgebra is maximal toral if it is maximal by inclusion among toral subalgebras of $\mathfrak{g}$.