The existence of the $[r, s, n]$ sum of square formula:
$(x_1^2+ \ldots +x_r^2) \cdot (y_1^2+ \ldots +y_s^2) = (z_1^2+ \ldots +z_n^2)$
is related to the existence of an axial map of projective spaces:
$P^{r - 1} \times P^{s-1} \to P^{n-1}$
There is a recent work extending this formula to some fields of non-zero characteristic:
