In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible.
of a finite-dimensional Lie algebra
is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent.
It is an ideal in the radical
of the Lie algebra
The quotient of a Lie algebra by its nilradical is a reductive Lie algebra
However, the corresponding short exact sequence does not split in general (i.e., there isn't always a subalgebra complementary to
This is in contrast to the Levi decomposition: the short exact sequence does split (essentially because the quotient