Cubic sublattices

A sublattice of the three-dimensional integer lattice $\mathbb Z^3$ is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector $\mathbf v\in\mathbb Z^3$ whose squared length is divisible by $d^2$, there exists a cubic sublattice containing $\mathbf v$ with edge length $d$. This improves one of the main result of a paper [arxiv:0806.3943] of Goswick et al., where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices.