  A solvable Lie algebra L has the property that its nilradical N contains its
own centraliser. This is interesting because 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), where Z(N) is the centre of the
nilradical of L, is isomorphic to a subalgebra of Der(N?*) where N*? is an
ideal of L such that N?*/N is the socle of a semisimple Lie algebra.
