  Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras and
$\eta$ be a non-zero complex number. A nonlinear bijective map $\phi:\mathcal
A\rightarrow\mathcal B$ has been demonstrated to satisfy
$$\phi([A,B]_{*}^{\eta}\diamond_{\eta}
C)=[\phi(A),\phi(B)]_{*}^{\eta}\diamond_{\eta}\phi(C)$$ for all
$A,B,C\in\mathcal A.$ If $\eta=1,$ then $\phi$ is a linear $*$-isomorphism, a
conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or
the negative of a conjugate linear $*$-isomorphism. If $\eta\neq 1$ and
satisfies $\phi(I)=1,$ then $\phi$ is either a linear $*$-isomorphism or a
conjugate linear $*$-isomorphism.
