  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 $\mathbb{Z}^4$.
