Unitary groups of nondegenerate Hermitian forms and the homotopy groups of pseudospheres

The Witt Extension Theorem states that the unitary group of a finite-dimensional vector space V equipped with a nondegenerate hermitian form acts transitively on the pseudosphere induced by the form. We provide a new, constructive proof of this result for finite-dimensional vector spaces V over R, C, or H. This constructive proof is then used to prove a similar result for the unitary group of a finitely generated free right module over an abelian AW^∗-algebra. The topology of these unitary groups is examined and as an application we determine the homotopy groups π₁ and π₂ of the induced real, complex, and quaternionic pseudospheres.