Associated with a non-degenerate symmetric bilinear form on a vector space is a Clifford algebra and various Clifford groups, which have spin representations on minimal right ideals of the Clifford algebra. Several invariants for these representations have been known for some time. In this paper the forms are assumed to be “split”, and several relations between the invariants are derived and promoted to the status of axioms. Then it is shown that any system satisfying the axioms comes from a minimal right ideal in a Clifford algebra and that the automorphism groups associated with the system are the Clifford groups. Hence, the axioms characterize spin representations.

A description of split forms and spin representations is in section two. In section three the invariants and their properties are described.