Extensions and automorphisms of Lie algebras

Let 0 -> A -> L -> B -> 0 be a short exact sequence of Lie algebras over a field F, where A is abelian. We show that the obstruction for a pair of automorphisms in Aut(A) x Aut(B) to be induced by an automorphism in Aut(L) lies in the Lie algebra cohomology H-2(B; A). As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in Aut (L-n,2((1))) x Aut (L-n,2(ab)) to be induced by an automorphism in Aut (L-n,L-2 ), where (L-n,L-2) is a free nilpotent Lie algebra of rank n and step 2.