Let \({\mathcal {M}}\) be a finite von Neumann algebra with no central summands of type \(I_{1}\). Assume that \(\delta :{\mathcal {M}}\rightarrow {\mathcal {M}}\) is a nonlinear map satisfying \(\delta ([[A,B],C])=[[\delta (A),B],C]+[[A,\delta (B)],C]+[[A,B],\delta (C)]\) for any \(A,B,C\in {\mathcal {M}}\) with \(ABC=0\). Then, we prove that there exists an additive derivation \(d:{\mathcal {M}}\rightarrow {\mathcal {M}}\), such that \(\delta (A)=d(A)+\tau (A)\) for any \(A\in {\mathcal {M}}\), where \(\tau :{\mathcal {M}}\rightarrow {\mathcal {Z}}_{{\mathcal {M}}}\) is a nonlinear map, such that \(\tau ([[A,B],C])=0\) for any \(A,B,C\in {\mathcal {M}}\) with \(ABC=0\).

令\（{\ mathcal {M}} \）是一个不具有\（I_ {1} \）类型中心求和的有限冯诺依曼代数。假设\（\ delta：{\ mathcal {M}} \ rightarrow {\ mathcal {M}} \）是一个满足\（\ delta（[[A，B]，C]）= [[\ delta （A），B]，C] + [[A，\ delta（B）]，C] + [[A，B]，\ delta（C）] \）对于任何\（A，B，C \ in {\ mathcal {M}} \）和\（ABC = 0 \）。然后，我们证明存在一个加法导数\（d：{\ mathcal {M}} \ rightarrow {\ mathcal {M}} \\），使得\（\ delta（A）= d（A）+ \ tau （A）\）对于任何\（A \ in {\ mathcal {M}} \\），其中\（\ tau：{\ mathcal {M}} \ rightarrow {\ mathcal {Z}} _ {{\ mathcal { M}}} \）是一个非线性映射，对于{\ mathcal {M}} \\中的任何\（A，B，C \\）和\（ABC是\\ tau（[[A，B]，C]）= 0 \）= 0 \）。