TITLE: seeking an integer parameterization for A^2+B^2=C^2+D^2+1
QUESTION [6 upvotes]: I'm looking for a complete [integer] parameterization of all integer solutions to the Diophantine equation
$A^2+B^2=C^2+D^2+1$,
analogous to the classical parameterization of the Pythagorean equation, i.e.
$A^2+B^2=C^2 \implies t,m,n \text{ such that } (A,B,C)=t(m^2-n^2,2mn,m^2+n^2)$.
Dickson's History contains many references and examples, but most appear to be inadequate, incomplete, or simply incorrect. Barnett and Bradley independently reached almost the same parameterization of the more general equation
$A^2+B^2+C^2=D^2+E^2+F^2$,
but I have so far been unable to reduce their parameterization(s) to one which solves the first equation I posted.
Any help or further references would be greatly appreciated.
Thanks!
Kieren.

REPLY [5 votes]: Such a parametrization is not possible.
Proof. Suppose A,B,C,D are polynomials with integer coefficients (in any number of variables) and A^2+B^2 = C^2+D^2+1. 
Then we have a parametrization for the congruence subgroup  H of SL(2,Z) reducing to
the permutation matrices modulo 2 (see Noam D. Elkies Aug 13 '12 at 15:49 above).
This H appears in Example 14 in my Annals paper. The first rows in H are (A+C, B+D).
Modulo 2, (A+C)(B+D) = 0, hence either  A+C or C+D is always even. But those rows contain both (1,0) and (0,1). This is a contradiction. QED.