A Converse to the Whitehead Theorem

In this postscript to our earlier note (J. Lie Theory 18 (2008), 295-299) we show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and nilpotent algebras.