Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over a field of characteristic 0, with solvable radical $\mathfrak{r}$ and nilpotent radical $\mathfrak{n}=[\mathfrak{g},\mathfrak{r}]$. Given a finite dimensional $\mathfrak{g}$-module $U$, its nilpotency series $ 0\subset U({\mathfrak n}^1)\subset\cdots\subset U({\mathfrak n}^m)=U$isdefinedsothat$U({\mathfrak n}^1)$isthe0-weightspaceof${\mathfrak n}$in$U$,$U({\mathfrak n}^2)/U({\mathfrak n}^1)$isthe0-weightspaceof${\mathfrak n}$in$U/U({\mathfrak n}^1)$,andsoon\mathrm{.}Wesaythat$U$ is linked if each factor of its nilpotency series is a uniserial ${\mathfrak g}/{\mathfrak n}$-module,i\mathrm{.}e\mathrm{.},its${\mathfrak g}/{\mathfrak n}$-submodules form a chain. Every uniserial $\mathfrak{g}$-module is linked, every linked ${\mathfrak g}$-module is indecomposable with irreducible socle, and both converse fail. In this paper we classify all linked $\mathfrak{g}$-modules when ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$and$$\mathrm{a}\mathrm{d}$, x$ acts diagonalizably on the abelian Lie algebra $\mathfrak{a}$. Moreover, we identify and classify all uniserial $\mathfrak{g}$-module amongst them.