We characterise the modules B of homological type FP,, over a finitely generated Lie algebra L such that L is a split extension of an abelian ideal A and an abelian subalgebra Q and A acts trivially on B. The characterisation is in terms of the invariant A introduced by R. Bryant and J. Groves and is a Lie algebra version of the generalisation (K 4, Conjecture 1] of the still open FPm-Conjecture for metabelian groups [Bi-G, Conjecture p. 367]. The case m = 1 of our main result is treated separately, as there the characterisation is proved without restrictions on the type of the extension.129221239