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