... completes the proof. HIGHER COMPOSITION LAWS I243Again, the proof gives very precise information on the bijection of The- orem 17. Given a pair (S, (I,M)), the corresponding pair of 4 × 4 skew-symmetric ... triples(I1,I2,I3) of a quadratic ring S, the case of pairs of binary quadratic forms canbe handled by imposing a twofold symmetry. The method of proof is similar;we simply state the result.Theorem 16. There ... from this composition lawon2× 2 × 2 cubes four further new laws of composition. These laws of composition are defined on 1) binary cubic forms, 2) pairs of binary quadraticforms, 3) pairs of quaternary...