Some remarks on Spin-orbits of unit vectors

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