Some remarks on spin-orbits of unit vectors

For $n\in N$ and a commutative ring R with $2\in R^{\times }$, the group $S{L}_n\left(R\right)$ acts on the set $U{m}_n\left(R\right)$ of unimodular vectors of length n and $Spi{n}_{2n}\left(R\right)$ acts on the set of unit vectors $U_{2n-1}\left(R\right)$. We give an example of a ring for which the comparison map $U{m}_n\left(R\right)/S{L}_n\left(R\right)\to U_{2n-1}\left(R\right)/Spi{n}_{2n}\left(R\right)$ fails to be bijective.