I've seen a couple of constructions of the so-called Witt group: it seems that most authors start with the commutative monoid of isometry classes of quadratic spaces under direct sum, pass to the Grothendieck group, and then quotient out the subgroup generated by the hyperbolic plane (sometimes called the split quadratic spaces).
In Bump's Automorphic Forms and Representations, however, it is claimed that one need only quotient out the submonoid generated by the hyperbolic plane and the monoid thus obtained is already a group. So we can avoid the Grothendieck construction entirely. The content of this statement is that given a quadratic space, there exists another such that the direct sum of the two is split (i.e. a sum of copies of the hyperbolic plane). But I don't see why. Does anyone have a reference or an explanation?