of such vector bundles is much more difficult to compute. At the moment, this is known only for the circle $S^1$ (obvious) and the two-dimensional sphere $S^2$ ([@BO]). The result on $S^3$ could be thought of as a measure of the recent progresses made in understanding the unstable structure of projective modules.

The paper is organized as follows: In Section \[symplectic\], we recall a few basic facts about symplectic modules. This includes the definitions of the group $\K A$, and the definition of the Euler class of a projective module of rank $2$. The following section is devoted to the study of the set of unimodular rows of even length $n$ under the action of the symplectic group $Sp_n(A)$. We show that a unimodular row of even length $n$ gives a stably free module of rank $n-2$ endowed with an antisymmetric form. This form is stably isometric to $H(A^{n/2})$ (whose class in $\K A$ is therefore trivial) and becomes isometric to $H(A^{(n-2)/2})$ if and only if its associated unimodular row is in the $Sp_{n}(A)$-orbit of $(1,0,\ldots,0)$. We prove the theorem on the freeness of projective modules of rank $2$ in Section \[core\]. The basic (and classical) observation here is that 