Splittable Lattices in the metabelian solvable Lie group ${\mathbb{R}}^n⋊{\mathbb{R}}^m$

The purpose of this note is describe and classify the splittable lattices in the completely solvable metabelian Lie group (semidirect product of abelian vector groups) $G:={\mathbb{R}}^n{⋊}_{\eta }{\mathbb{R}}^m$, where $\eta$ is the continuous representation of the topological additive abelian group ${\mathbb{R}}^m$ in ${\mathbb{R}}^n$ given by $\eta (t_1,\dots ,t_m)=\exp ({\sum }_{j=1}^m{t}_j{\mathrm{\Delta }}_j)$ with $({\mathrm{\Delta }}_j{)}_{1\le j\le m}$ is a set of pairwise commuting diagonal matrices in ${\mathbb{R}}^{n\times n}$.