Question: What is the correct notion of a product of integral (or rational) polytopes which induces a factorization of its Ehrhart (quasi-)polynomial into two primitive Ehrhart (quasi-)polynomials corresponding to its constituent polytopes, viz., $L_{P \times Q}(t) = L_{P}(t) L_{Q}(t)$?
(Motivation) Given two closed integral polytopes $P$ and $Q$ each with vertices at $\{ \mathbf{0} , b_{1} \mathbf{e}_{1}, \dots, b_{n} \mathbf{e}_{n} \}$ and $\{ \mathbf{0} , d_{1} \mathbf{e}_{1}, \dots, d_{m} \mathbf{e}_{m} \}$, respectively, where $n, b_{i}, m, d_{j} \in \mathbb{N}$, define the integral polytope $R$ with vertices at $\{ \mathbf{0}, b_{1} \mathbf{e}_{1}, \dots, b_{n} \mathbf{e}_{n}, d_{1} \mathbf{e}_{n+1}, \dots, d_{m} \mathbf{e}_{n+m} \}$.
The above construction cannot be the sought after product $P \times Q$. Suppose $P$ and $Q$ are defined by $b_{1} = b_{2} = d_{1} = d_{2} = 2$. It is easy to show that $L_{P}(1) = L_{Q}(1) = 6$. Define $R$ as above with vertices of $P$ and $Q$. It is true that $L_{R}(1) = 15 \neq 6^{2}$.
Question: What is $R$ in terms of $P$ and $Q$? Is it special in some way?
Thanks!