  We study the root polytope $\mathcal P_\Phi$ of a finite irreducible
crystallographic root system $\Phi$ using its relation with the abelian ideals
of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We
determine the hyperplane arrangement corresponding to the faces of codimension
2 of $\mathcal P_\Phi$ and analyze its relation with the facets of $\mathcal
P_\Phi$. For $\Phi$ of type $A_n$ or $C_n$, we show that the orbits of some
special subsets of abelian ideals under the action of the Weyl group
parametrize a triangulation of $\mathcal P_\Phi$. We show that this
triangulation restricts to a triangulation of the positive root polytope
$\mathcal P_\Phi^+$.
