On the transposition anti-involution in real Clifford algebras III: the automorphism group of the transposition scalar product on spinor spaces

In Ablamowicz and Fauser [R. Ablamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras I: The transposition map, Linear Multilinear Alg. (to appear)] a signature epsilon = (p, q)-dependent transposition anti-involution T-epsilon(similar to) of real Clifford algebras Cl-p,Cl-q for non-degenerate quadratic forms was introduced. In Ablamowicz and Fauser [R. Ablamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Alg. (to appear)] we showed that, depending on the value of (p - q) mod 8, the map T-epsilon(similar to) gives rise to transposition, complex Hermitian or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [R. Ablamowicz and B. Fauser, Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple, Comput. Phys. Commun. 170 (2005), pp. 115-130]. We provide a full signature (p, q)- dependent classification of the invariance groups G(p,q)(epsilon) of this product for p + q <= 9. The map T-epsilon(similar to) is identified as the 'star' map known [D. S. Passman, The Algebraic Structure of Group Rings, Robert E. Krieger Publishing Company, Malabar, Florida, 1985] from the theory of (twisted) group algebras where the Clifford algebra Cl-p,Cl-q is seen as a twisted group ring R-t[(Z(2))(n)], n = p + q. We discuss important subgroups of a stabilizer group G(p,q)(f) of a primitive idempotent f and we relate their transversals to spinor bases in spinor spaces realized as minimal left ideals Cl(p,q)f.