# 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 (i.e., there isn't always a *subalgebra* complementary to in ). This is in contrast to the Levi decomposition: the short exact sequence

does split (essentially because the quotient is semisimple).

## See also

- Levi decomposition
- Nilradical of a ring, a notion in ring theory.

## References

- Fulton, William; Harris, Joe (1991).
*Representation theory. A first course*. Graduate Texts in Mathematics, Readings in Mathematics.**129**. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249, ISBN 978-0-387-97527-6. - Onishchik, Arkadi L.; Vinberg, Ä–rnest Borisovich (1994),
*Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras*, Springer, ISBN 978-3-540-54683-2.

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