Beitraege zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 34 (1993), No. 1, 15-30. Separation in the Polytope Algebra Peter McMullen Abstract. The polytope algebra is the universal group for translation invariant valuations on the family of polytopes in a finite dimensional vector space over an ordered field. In an earlier paper, it was shown that the polytope algebra is, in all but one trivial respect, a graded (commutative) algebra over the base field. Also described was a family of separating (group) homomorphisms, called frame functionals. However, various questions relating to the frame functionals were left open, such as what syzygies exist between them, and what the image of a certain closely related mapping is. Here, these questions are settled: essentially, the only restrictions are imposed by the Minkowski relations. In doing this, simpler proofs are also found of some results in that earlier paper. Finally, there are consequences for expressing certain translation invariant valuations in terms of mixed volumes. MSC 1991: Primary 52B45; dissections and valuations.