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