I'm having trouble following what exactly the theorem is that you're citing. Here's what I am reading: "Given a degree-8 polynomial with roots , , , and , and given a number A, there exists a B and C such that if 2A is expressible as the sum of two squares in at least two unique ways." Is this correct?
This seems inconsistent. By expanding both sides of the equation, it is clear that given a,b,c,d, there exists one and only one combination A,B,C making this equation work, found by:
The algebra is long but elementary, and I have checked this with the raw data you provided, a=1,b=11,c=13,d=7, and indeed A=85,B=4176,C=2880. In other words, you cannot choose A - it is dependent on a,b,c,d. Is the proper wording of the theorem, "If 2A is expressible as the sum of two squares in two unique ways, then there exist integers B and C..."
I think your overall question can be answered by analyzing the proof of the theorem you are citing, and trying to generalize it.