Characterizations of Lie derivations of B(X)B(X)

Let XX be a Banach space of dimension greater than 2. We prove that if δ:B(X)→B(X)δ:B(X)→B(X) is a linear map satisfyingδ([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) with AB=0AB=0 (resp. AB=PAB=P, where PP is a fixed nontrivial idempotent), then δ=d+τδ=d+τ, where dd is a derivation of B(X)B(X) and τ:B(X)→CIτ:B(X)→CI is a linear map vanishing at commutators [A,B][A,B] with AB=0AB=0 (resp. AB=PAB=P).