A 4-dimensional cross-polytope (also called 16-cell) is a regular polytope whose vertices are all permutations of $(\pm1,0,0,0)$. It is known that this body tiles the space $\mathbb{R}^4$ by translation and that the set of translation vectors describing the tiling can be chosen to form a lattice $L$.

My question is how can we determine the generator matrix of $L$ (or give some equivalent explicit description)?

If the answer can be given for general lattice-tiles, that would be even better of course.

**EDIT:** The statement that 16-cell is a lattice-tile is incorrect. I apologize for this. See the answer at: https://math.stackexchange.com/q/1923688/292993.