Recall that a sum of squares formula for $[r,s,n]$ over a field $F$ is an identity of the form $$ ( x_{1}^{2} + \cdots + x_{r}^{2})( y_{1}^{2} + \cdots + y_{s}^{2}) = ( z_{1}^{2} + \cdots + z_{n}^{2}),$$ where the $z_{k}$ are bilinear expressions in the $x_{i}$ and $y_{j}$. The simplest non-trivial example is $$(x_{1}^{2} + x_{2}^{2})(y_{1}^{2} + y_{2}^{2}) = (x_{1}y_{1} - x_{2}y_{2})^{2} + (x_{1}y_{2} + x_{2}y_{1})^{2}$$ coming from the complex numbers. Similar examples can be given from the multiplication on the quaternions and octonions.
Various conditions for the existence of sums of squares formulae are know, but there seem to be very few examples of such formulae. Are there any known algorithms for generating identities? What about in the special cases where identities are known to exist? For example, the case $[r,n,n]$ was solved by Hurwitz and Radon (independently). There exists a sum of squares formula with signature $[r,n,n]$ if and only if $r = \rho(n)$, where rho is the Radon--Hurwitz number. Are these identities constructible? The theorem indicates the existence of an identity with signature $(9,48,48)$ for example. How might we go about finding these?
