The existence of the [r, s, n] sum of square formula:
(x1^2+, ...+xr^2)*(y1^2+, ...+ys^2) = (z1^2+, ...+xn^2)
is related to the existence of an axial map of projective spaces:
P^(r-1)xP^(s-1)-->P^(n-1)
There is a recent work extending this formula to some fields of non-zero characteristic: