  The quadric $\operatorname{Q}_{2n}$ is the ${\mathbb Z}$-scheme defined by
the equation $\sum_{i=1}^n x_i y_i = z(1-z)$. We show that
$\operatorname{Q}_{2n}$ is a homogeneous space for the split reductive group
scheme $\operatorname{SO}_{2n+1}$ over ${\mathbb Z}$. The quadric
$\operatorname{Q}_{2n}$ is known to have the ${\mathbb A}^1$-homotopy type of a
motivic sphere and the identification as a homogeneous space allows us to give
a characteristic independent affine representability statement for motivic
spheres. This last observation allows us to give characteristic independent
comparison results between Chow--Witt groups, motivic stable cohomotopy groups
and Euler class groups.
