Polar Root Polytopes that are Zonotopes

Let ${\mathcal{P}}_{\mathrm{\Phi }}$ be the root polytope of a finite irreducible crystallographic root system $\mathrm{\Phi }$, i.e., the convex hull of all roots in $\mathrm{\Phi }$. The polar of ${\mathcal{P}}_{\mathrm{\Phi }}$, denoted ${\mathcal{P}}_{\mathrm{\Phi }}^{\ast }$, coincides with the union of the orbit of the fundamental alcove under the action of the Weyl group. In this paper, we establishes which polytopes ${\mathcal{P}}_{\mathrm{\Phi }}^{\ast }$ are zonotopes and which are not. The proof is constructive.