Triangulations of root polytopes

Let $\mathrm{\Phi }$ be an irreducible crystallographic root system and $\mathcal{P}$ its root polytope, i.e., its convex hull. We provide a uniform construction, for all root types, of a triangulation of the facets of $\mathcal{P}$. We also prove that, on each orbit of facets under the action of the Weyl gruop, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.