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Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras and$\eta$ be a non-zero complex number. A nonlinear bijective map $\phi:\mathcalA\rightarrow\mathcal B$ 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, aconjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, orthe negative of a conjugate linear $*$-isomorphism. If $\eta\neq 1$ andsatisfies $\phi(I)=1,$ then $\phi$ is either a linear $*$-isomorphism or aconjugate linear $*$-isomorphism.

Nonlinear maps preserving the mixed Jordan triple $\eta$-$*$-product between factors