TITLE: What does Pfister's 8-Square Identity look like?
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: Here's Maple code for getting a 16-square version using the method Conrad outlines.  The result is too big to show here (the equation eq has length 4808351).

C2:= < < c[1] | c[2] >,<-c[2] | c[1]> >:
c2:= c[1]^2 + c[2]^2:
C2b:= subs(seq(c[i]=c[i+2],i=1..2),C2):
c4:= add(c[i]^2,i=1..4):
C4:= < < C2|C2b >,< -C2 . C2b^%T . C2/c2 | C2 > >:
C4b:= subs(seq(c[i]=c[i+4],i=1..4),C4):
c8:= add(c[i]^2,i=1..8):
C8:= < < C4|C4b >,< -C4 . C4b^%T . C4/c4 | C4 > >:
C8b:= subs(seq(c[i]=c[i+8],i=1..8),C8):
C16:= < < C8|C8b >,< -C8 . C8b^%T . C8/c8 | C8 > >:
XY:= subs(c=x, C16) . subs(c=y, C16):
for i to 16 do z[i] := XY[1, i] end do:

To verify:

eq:= add(x[i]^2, i=1..16) * add(y[i]^2, i=1..16) = add(z[i]^2, i=1..16):
testeq(eq);

true