Nilradical of a Lie algebra

In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible.

The nilradical $$\mathfrak{nil}(\mathfrak g)$$ of a finite-dimensional Lie algebra $$\mathfrak{g}$$ is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical $$\mathfrak{rad}(\mathfrak{g})$$ of the Lie algebra $$\mathfrak{g}$$. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra $$\mathfrak{g}^{\mathrm{red}}$$. However, the corresponding short exact sequence
 * $$ 0 \to \mathfrak{nil}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{red}}\to 0$$

does not split in general (i.e., there isn't always a subalgebra complementary to $$\mathfrak{nil}(\mathfrak g)$$ in $$\mathfrak{g}$$). This is in contrast to the Levi decomposition: the short exact sequence
 * $$ 0 \to \mathfrak{rad}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{ss}}\to 0$$

does split (essentially because the quotient $$\mathfrak{g}^{\mathrm{ss}}$$ is semisimple).