  Let $\mathcal P_{\Phi}$ be the root polytope of a finite irreducible
crystallographic root system $\Phi$, i.e., the convex hull of all roots in
$\Phi$. The polar of $\mathcal P_{\Phi}$, denoted $\mathcal P_{\Phi}^*$,
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
$\mathcal P_{\Phi}^*$ are zonotopes and which are not. The proof is
constructive.
