  In this paper, we study flag structures of matroid base polytopes. We
describe faces of matroid base polytopes in terms of matroid data, and give
conditions for hyperplane splits of matroid base polytopes. Also, we show how
the cd-index of a polytope can be expressed when a polytope is split by a
hyperplane, and apply these to the cd-index of a matroid base polytope of a
rank 2 matroid.
