In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible.
The nilradical n i l ( g ) of a finite-dimensional Lie algebra 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 r a d ( g ) of the Lie algebra g . The quotient of a Lie algebra by its nilradical is a reductive Lie algebra g r e d . However, the corresponding short exact sequence
0 → n i l ( g ) → g → g r e d → 0 does not split in general (i.e., there isn't always a subalgebra complementary to n i l ( g ) in g ). This is in contrast to the Levi decomposition: the short exact sequence
0 → r a d ( g ) → g → g s s → 0 does split (essentially because the quotient g s s is semisimple).
