Two lives: Compositions of unimodular rows

The paper lays the foundation for study of unimodular rows using Spin groups. We show that elementary orbits (of any length n≥3) are equivalent to on unit sphere. (This bijection is proved over all commutative rings.) In special case n=3, this leads an interpretation Vaserstein symbol addition, we introduce a new composition law operates certain subspaces underlying quadratic space (using multiplication in algebras). particular, split-quaternions (discovered by L. and later generalized W. van der Kallen). Strikingly, with approach, now see possibility orbit structures not only octonion multiplication) but also more general spaces.