By (r, s, t)-identity I mean any sort of such identity:
$$ 
(x\_1^2+\ldots + x\_r^2)(y\_1^2+\ldots +y\_s^2)=(z\_1(x,y)^2+\ldots + z\_t(x,y)^2),
$$
where $z\_i(x,y)$ is a polynomial for every $i$.


See this for some further reading: <https://en.wikipedia.org/wiki/Hurwitz_problem> 


I can't find in the web any non-standard examples of such identities like Adem's example of (3,5,7)-identity or some others for small r, s and t. Can anyone provide several such expressions?



On Hurwitz Square (r, s, t)-Identities examples
From pages 137-138 of [Rajwade](http://oskicat.berkeley.edu/record=b14952300~S1), "Notes on Chapter 10"


the Hurwitz-Radon theorem gives
$$ (2,2,2), \; (4,4,4), \; (8,8,8), \; (9,16,16), \; (10,32,32), \ldots $$ 


K. Y. Lam found $(10,10,16)$ in 1966.


In 1975, Adem found
$$ (3,5,7), \; (10,10,16), \; (12,12,28), \; (13,13,28), \; (17,18,32), \ldots $$


In 1982, Yuzvinski found
$$ (10,10,16), \; (12,20,32), \; (14,40,64), \ldots $$


He recommends a 1989 survey article by Shapiro called *Products of Sums of Squares*


<https://people.math.osu.edu/shapiro.6/lec1.pdf> 


<https://people.math.osu.edu/shapiro.6/lec2.pdf>


<https://people.math.osu.edu/shapiro.6/lec3.pdf>


