Nilradical of a Lie algebra

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In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible.

The nilradical \mathfrak{nil}(\mathfrak g) of a finite-dimensional Lie algebra \mathfrak{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 \mathfrak{rad}(\mathfrak{g}) of the Lie algebra \mathfrak{g}. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra \mathfrak{g}^{\mathrm{red}}. However, the corresponding short exact sequence

 0 \to \mathfrak{nil}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{red}}\to 0

does not split in general (i.e., there isn't always a subalgebra complementary to \mathfrak{nil}(\mathfrak g) in \mathfrak{g}). This is in contrast to the Levi decomposition: the short exact sequence

 0 \to \mathfrak{rad}(\mathfrak g)\to \mathfrak g\to \mathfrak{g}^{\mathrm{ss}}\to 0

does split (essentially because the quotient \mathfrak{g}^{\mathrm{ss}} is semisimple).

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