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

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) $G:=\mathbb{R}^n\rtimes_\eta\mathbb{R}^m$, where $\eta$ is the continuous representation of the topological additive abelian group $\mathbb R^m$ in $\mathbb R^n$ given by $\eta(t_1,\dots, t_m)=\exp(\sum_{j=1}^{m}t_j\Delta_j)$ with $(\Delta_j)_{1\leq j\leq m}$ is a set of pairwise commuting diagonal matrices in $\mathbb R^{n\times n}$.