Let B be a ∗-algebra with the unity and a nontrivial projection. In the present paper, we show under certain restrictions that if a map Ψ : B →B satisfies Ψ([E ⋄ K, D]∗) = [Ψ(E) ⋄ K, D]∗ + [E ⋄Ψ(K), D]∗ + [E ⋄ K, Ψ(D)]∗ for all E, K, D ∈ B, then Ψ is an additive ∗-derivation.

Let $ \mathfrak{A} $ be a unital $ * $-algebra with identity $ \mathscr{I} $. For $ \mathscr{A}, \mathscr{B}\in\mathfrak{A} $, define the bi-skew Jordan product $ \mathscr{A}\bullet \mathscr{B} = \mathscr{A}\mathscr{B}^{*}+\mathscr{B}\mathscr{A}^{*} $ and the bi-skew Lie product $ [\mathscr{A},\mathscr{B}]_{\diamond} = \mathscr{A}\mathscr{B}^{*}-\mathscr{B}\mathscr{A}^{*}. $ Suppose that a nonlinear mapping $ \Phi:\mathfrak{A}\to\mathfrak{A} $ satisfies $ \Phi\!\left(\big[\mathscr{A}_1\bullet\mathscr{A}_2\bullet\cdots\bullet\mathscr{A}_{n-1},\mathscr{A}_n\big]_{\diamond}\right) = \sum\limits_{j = 1}^{n}\big[\mathscr{A}_1\bullet \cdots \bullet\mathscr{A}_{j-1}\bullet \Phi(\mathscr{A}_j)\bullet\mathscr{A}_{j+1}\bullet \cdots\bullet\mathscr{A}_{n-1},\mathscr{A}_n\big]_{\diamond} $ for all suitable elements $ \mathscr{A}_1, \ldots, \mathscr{A}_n\in\mathfrak{A} $, where $ \mathscr{A}_1, \mathscr{A}_2\in\{\mathscr{I}, i\mathscr{I}\} $ and $ \mathscr{A}_j = \mathscr{I} $ for every $ j = 3, 4, \ldots, n-2 $. We prove that $ \Phi $ is an additive $ * $-derivation on $ \mathfrak{A} $. As applications, several consequences are obtained for prime $ * $-algebras, factor von Neumann algebras, von Neumann algebras without central summands of type $ \mathscr{I}_1 $, and standard operator algebras. Moreover, a conjecture is proposed to motivate further research in this direction. The obtained results extend and generalize a recent result of Abbasi et al. [Non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $ * $-algebras, Ricerche Mat., 2026.] concerning non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on $ * $-algebras.

Consider $\mathcal{A}$ as a unital ∗-algebra. Given elements $A,B\in \mathcal{A}$, the operations $A•B=AB+B{A}^{*}$ and ${[A,B]}_{*}=AB-B{A}^{*}$ represent the skew Jordan product and the skew Lie product, correspondingly. Within this study, we demonstrate that a nonlinear mapping $\mathrm{\Phi }:\mathcal{A}\to \mathcal{A}$ adhering to $\mathrm{\Phi }\left({[A_1•A_2•\cdots •A_{n-1},A_n]}_{*}\right)={\sum }_{j=1}^n{[A_1•A_2•\cdots •A_{j-1}•\mathrm{\Phi }(A_j)•A_{j+1}•\cdots •A_{n-1},A_n]}_{*}$ for every $A_1,A_2,\dots ,A_n\in \mathcal{A}$ with $n\ge 3$, then $\mathrm{\Phi }$ is an additive ∗-derivation. As an application of our main results, we establish characterizations of prime ∗-algebras, standard operator algebras, von Neumann algebras without central summands of type ${\mathrm{I}}_1$, and factor von Neumann algebras.

Let R be a 2-torsion free commutative unital ring and I(S, R) be the incidence algebra of a locally finite preordered set S. In the present paper, we show that if an R-linear map T: I(S, R) ? I(S, R) satisfies T([t ? u, v]) = [T(t) ? u, v] + [t ? T(u), v] + [t ? u, T(v)], for all t, u, v ? I(S, R), then T = ? + ?, where ?: I(S, R) ? I(S, R) is a derivation and ?: I(S, R) ? Z(I(S, R)) is an R-linear map.