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).
