Let U be a von Neumann algebra endowed with the Lie product [A,B] = AB − BA (A,B ∈ U ). In this article, we consider the subsequent condition on an additive mapping φ on the von Neumann algebra U with a suitable projection P ∈ U : φ([[A,B],C]) = [[φ(A),B],C] = [[A,φ(B)],C] for all A,B,C ∈ U with AB = P and we show that φ(A) = WA + ξ(A) for all A ∈ U , where W ∈ Z(U ), and ξ : U → Z(U ) (Z(U ) is the center of U ) is an additive map in which ξ([[A,B],C]) = 0 for any A,B,C ∈ U with AB = P. We also give some results of the conclusion.
