On cohomologies and representations of groups with normal Engel subgroups

Let $\lambda$, $U$ be representations of a group $G$ with a normal Engel subgroup $N$. The paper studies triviality conditions for the cohomology group $\mathcal{H}^1(G,\lambda,U)$ when $\lambda$ and $U$ are sectionally spectrally disjoint and examines some decompositions of the extension $\mathfrak{e}(\lambda, U,\xi)$ of $\lambda$ by $U$ associated with non-trivial $(\lambda, U)$-cocycles $\xi$.