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

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Let g be a finite dimensional Lie algebra over a field of characteristic 0, with solvable radical r and nilpotent radical n=[g,r]. Given a finite dimensional g-module U, its nilpotency series 0⊂ U(n¹)⊂⊂ U(nᵐ)=U is defined so that U(n¹) is the 0-weight space of n in U, U(n²)/U(n¹) is the 0-weight space of n in U/U(n¹), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial g/n-module, i.e., its g/n-submodules form a chain. Every uniserial g-module is linked, every linked g-module is indecomposable with irreducible socle, and both converse fail. In this paper we classify all linked g-modules when g= x a and ad x acts diagonalizably on the abelian Lie algebra a. Moreover, we identify and classify all uniserial g-module amongst them.