We are familiar with Hurwitz’s theorem which implies there is only the Fibonacci 2-Square, Euler 4-Square, Degen 8-Square, and no more. However, if we relax conditions and allow for rational expressions, then Pfister's theorem states that similar identities are possible for ALL $2^n$ squares. His 4-square version is:
$\begin{align}&(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\\ &(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1)^2 +\\ &(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2)^2 +\\ &\left(a_1 b_2 + a_2 b_1 + \frac{a_3 (b_1^2b_4-2b_1b_2b_3-b_2^2b_4)}{b_1^2+b_2^2} - \frac{a_4 (b_1^2b_3+2b_1b_2b_4-b_2^2b_3)}{b_1^2+b_2^2}\right)^2+\\ &\left(a_1 b_1 - a_2 b_2 - \frac{a_4 (b_1^2b_4-2b_1b_2b_3-b_2^2b_4)}{b_1^2+b_2^2} - \frac{a_3 (b_1^2b_3+2b_1b_2b_4-b_2^2b_3)}{b_1^2+b_2^2}\right)^2 \end{align}$
Question: What does the Pfister 8-square version look like? (And, if you have the strength, can you also give the 16-square version?) Here is K. Conrad's pdf file which describes the general method, but I can’t make heads or tails out of it.
$\color{red}{\text{Attention}}$ (Feb. 16): Someone is trying to delete Wikipedia's article on Degen's Eight-Square Identity simply because he finds it uninteresting. Please vote to keep.