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Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ isthe irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as anabelian Lie algebra. It is known that the isomorphism classes of uniserial$\mathfrak{g}_m$-modules consist of a family, say of type $Z$, containingmodules of arbitrary composition length, and some exceptional modules withcomposition length $\le 4$.  Let $V$ and $W$ be two uniserial $\mathfrak{g}_m$-modules of type $Z$. Inthis paper we obtain the $\mathfrak{sl}(2)$-module decomposition of$\text{soc}(V\otimes W)$ by giving explicitly the highest weight vectors. Itturns out that $\text{soc}(V\otimes W)$ is multiplicity free. Roughly speaking,$\text{soc}(V\otimes W)=\text{soc}(V)\otimes \text{soc}(W)$ in half of thecases, and in these cases we obtain the full socle series of $V\otimes W$ byproving that $ \text{soc}^{t+1}(V\otimes W)=\sum_{i=0}^{t}\text{soc}^{i+1}(V)\otimes \text{soc}^{t+1-i}(W)$ for all $t\ge0$.  As applications of these results, we obtain for which $V$ and $W$, the spaceof $\mathfrak{g}_m$-module homomorphisms $\text{Hom}_{\mathfrak{g}_m}(V,W)$ isnot zero, in which case is 1-dimensional. Finally we prove, for $m\ne 2$, thatif $U$ is the tensor product of two uniserial $\mathfrak{g}_m$-modules of type$Z$, then the factors are determined by $U$. We provide a procedure to identifythe factors from $U$.

Tensor products and intertwining operators for uniserial representations of the Lie algebra $\mathfrak{sl}(2)\ltimes V(m)$