Cubes of integral vectors in dimension four

A system of $m$ nonzero vectors in ${\mathbb{Z}}^n$ is called an $m$-icube if they are pairwise orthogonal and have the same length. The paper describes $m$-icubes in ${\mathbb{Z}}^4$ for $2\le m\le 4$ using Hurwitz integral quaternions, counts the number of them with given edge length, and proves that unlimited extension is possible in ${\mathbb{Z}}^4$.