  We compute the set of facets of the polytope which is the convex hull of the
Coxeter groups $\mathsf{F}_4$ or $\mathsf{H}_4$: For the group $\mathsf{F}_4$
we found $2$ orbits of facets which contradicts previous results published in
\cite{birkhoff}. For the group $\mathsf{H}_4$ we found $1063$ orbits of facets
which provides a counterexample to the conjecture of \cite{birkhoff}.
