Characterization of Lie derivations on von Neumann algebras

Let M be any von Neumann algebra without central summands of type I1. For any scalar ξ, a characterization of any additive map L on M satisfies L(AB-ξBA)=L(A)B-ξBL(A)+AL(B)-ξL(B)A whenever AB=0 is given: there exists an additive derivation φ such that, (1) if ξ=1, then L=φ+f, where f is an additive map into the center vanishing on [A,B] with AB=0; (2) if ξ=0, then L(I)∈Z(M) and L(A)=φ(A)+L(I)A for all A; (3) if ξ is rational and ξ≠0,1, then L=φ; (4) if ξ is not rational, then φ(ξI)=ξL(I) and L(A)=φ(A)+L(I)A.