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 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. This is in contrast to the Levi decomposition: the short exact sequence
does split.
