Nilradical

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a characteristic-zero field. Its nilradical, denoted $\mathrm{nilrad}(\mathfrak{g})$, is the largest nilpotent ideal of $\mathfrak{g}$: it is nilpotent in the lower-central-series sense (Lower central series and nilpotent Lie algebras), is an ideal (Lie subalgebras, ideals, and center), and contains every nilpotent ideal.

The word “largest” includes an existence assertion. It is supplied by Existence and characteristicity of the nilradical in characteristic zero ↗, which proves that sums of nilpotent ideals are nilpotent in this setting and then uses finite dimensionality. Thus $\mathrm{nilrad}(0)=0$, and if $\mathfrak{g}$ itself is nilpotent then $\mathrm{nilrad}(\mathfrak{g})=\mathfrak{g}$.

The nilradical is not defined as the set of all $x$ for which ${\mathrm{ad}}_x$ is nilpotent: that set need not be a linear subspace.