Affine representability of quadrics revisited

The quadric ${\mathrm{Q}}_{2n}$ is the $\mathbb{Z}$-scheme defined by the equation ${\sum }_{i=1}^n{x}_i{y}_i=z(1-z)$. We show that ${\mathrm{Q}}_{2n}$ is a homogeneous space for the split reductive group scheme ${\mathrm{SO}}_{2n+1}$ over $\mathbb{Z}$. The quadric ${\mathrm{Q}}_{2n}$ 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.