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

${\displaystyle 0\to {\mathfrak {nil}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}^{\mathrm {red} }\to 0}$

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

${\displaystyle 0\to {\mathfrak {rad}}({\mathfrak {g}})\to {\mathfrak {g}}\to {\mathfrak {g}}^{\mathrm {ss} }\to 0}$

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