Bounded common fundamental domains for two lattices

We prove that for any two lattices $L,M\subseteq {\mathbb{R}}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E\subseteq {\mathbb{R}}^d$ such that $E$ tiles ${\mathbb{R}}^d$ when translated by $L$ or by $M$. In fact, the set $E$ can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of $E$ forms a Weyl–Heisenberg (Gabor) orthogonal basis of $L^2({\mathbb{R}}^d)$ when translated by $L$ and modulated by $M^{\ast }$, the dual lattice of $M$.