  We study the cancellation property of projective modules of rank $2$ with a
trivial determinant over Noetherian rings of dimension $\leq 4$. If $R$ is a
smooth affine algebra of dimension $4$ over an algebraically closed field $k$
such that $6 \in k^{\times}$, then we prove that stably free $R$-modules of
rank $2$ are free if and only if a Hermitian $K$-theory group $\tilde{V}_{SL}
(R)$ is trivial.
