The generalised nilradical of a Lie algebra

A solvable Lie algebra L has the property that its nilradical N contains its own centraliser. This is interesting because it gives a representation of L as a subalgebra of the derivation algebra of its nilradical with kernel equal to the centre of N. Here we consider several possible generalisations of the nilradical for which this property holds in any Lie algebra. Our main result states that for every Lie algebra L  , L/Z(N)L/Z(N), where Z(N)Z(N) is the centre of the nilradical of L  , is isomorphic to a subalgebra of Der(N⁎)Der(N⁎) where N⁎N⁎ is an ideal of L   such that N⁎/NN⁎/N is the socle of a semisimple Lie algebra.