Tensor decomposition of a lattice (awaiting review)

Given integral lattices $L_1$ and $L_2$, we define the tensor product of $L_1$ and $L_2$ as the lattice $L = L_1 \otimes_{\Z} L_2$, where the associated bilinear form is defined as

$$ B( v_1 \otimes w_1, v_2 \otimes w_2 ) = B_{L_1} ( v_1, v_2) B_{L_2} (w_1, w_2), $$

extended bilinearly. The rank of $L$ is the product of the ranks of $L_1$ and $L_2$, and the determinant of $L$ is $ \det(L_1)^{\operatorname{rank}(L_2)} \cdot \det(L_2)^{\operatorname{rank}(L_1)}$.

We say that $L$ is tensor decomposable if it can be written as $L = L_1 \otimes L_2$ for some integral lattices $L_1$ and $L_2$. If not, we say $L$ is tensor indecomposable.

A tensor decomposition of $L$ is a decomposition of the form $L = L_1 \otimes L_2 \otimes \dots \otimes L_r$.