Extended form of $(\zeta ,\eta )$-derivation of von Neumann algebra

Let $\mathcal{U}$ be regarded as a CSL subalgebra of a von Neumann algebra operating on a Hilbert space $\mathcal{H}$. Let ${\mathcal{G}}_{\mathbf{h}},\mathbf{h}:\mathcal{U}\to \mathcal{U}$ denote two significant linear mappings that intricately fulfill certain essential algebraic identities. As a result, ${\mathcal{G}}_{\mathbf{h}}$ emerges as a generalized $(\zeta ,\eta )$-derivation, preserving a fundamental relationship with the associated $(\zeta ,\eta )$-derivation $\mathbf{h}$ functioning within the framework of $\mathcal{U}$.