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).