Lie derivable maps on B(X)B(X)

Let X   be a Banach space of dimension greater than 1. We prove that if a map δ:B(X)→B(X)δ:B(X)→B(X) satisfiesδ([A,B])=[δ(A),B]+[A,δ(B)]δ([A,B])=[δ(A),B]+[A,δ(B)] for any A,B∈B(X)A,B∈B(X), then δ=D+τδ=D+τ, where D   is an additive derivation of B(X)B(X) and the map τ:B(X)→FIτ:B(X)→FI vanishes at commutators [A,B][A,B].