A decomposition lemma for tetrahedra.
Any tetrahedron with side length λ+μ can be decomposed into two tetrahedra with side lenght λ and μ as well as two prisms. This is shows that the algebra of polyhedra modulo scissors congruence and prisms is indeed a real vector space. See “Jessen, B. (1968). The Algebra of Polyhedra and the Dehn-Sydler Theorem. Mathematica Scandinavica, 22, 241-256.” step 5.