Simple, semisimple, and reductive Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$.

• It is simple if it is nonabelian and its only ideals are $0$ and $\mathfrak{g}$.
• It is semisimple if $\mathrm{rad}(\mathfrak{g})=0$, as in Semisimple Lie algebras.
• When $\mathrm{char}k=0$, it is reductive when $\mathfrak{g}=Z(\mathfrak{g})\oplus [\mathfrak{g},\mathfrak{g}]$ and the derived algebra $[\mathfrak{g},\mathfrak{g}]$ is semisimple, as in Reductive Lie algebras.

The word “nonabelian” in the first clause excludes one-dimensional abelian Lie algebras from being simple. The zero Lie algebra is semisimple under the vanishing-radical convention and, when $\mathrm{char}k=0$, is reductive, with both displayed summands zero. These are conventions, not yet the structure theorem that every semisimple algebra is a direct sum of simple ideals.