Affine representability of quadrics revisited

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.