  We show that lattice polytopes cut out by root systems of classical type are
normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and
Trung in type A. We prove similar results for Cayley sums of collections of
polytopes whose Minkowski sums are cut out by root systems. The proofs are
based on a combinatorial characterization of diagonally split toric varieties.
