Reductive Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic zero. In this library, $\mathfrak{g}$ is reductive when

$\mathfrak{g}=Z(\mathfrak{g})\oplus [\mathfrak{g},\mathfrak{g}]$

as an internal direct sum of vector subspaces and the derived algebra $[\mathfrak{g},\mathfrak{g}]$ is semisimple in the vanishing-radical sense of Semisimple Lie algebras. Thus every element has a unique expression as a central element plus an element of the derived algebra; equivalently for the displayed sum, the two subspaces span $\mathfrak{g}$ and have zero intersection. The center is as in Lie subalgebras, ideals, and center, and the derived algebra is the first term after $\mathfrak{g}$ in Derived series and solvable Lie algebras.

The zero Lie algebra and every finite-dimensional abelian characteristic-zero Lie algebra are reductive under this convention: the derived algebra is zero, which is semisimple, and the center is all of $\mathfrak{g}$. Other standard characterizations of reductivity are not used here; their equivalence requires later structure theory. No choice principle is used.