In this MSE question/thread, I have been discussing the equation $$ (x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$} $$ where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to be false. Now, as an alternative, I'm looking for a "complete integer solution" (a.k.a. integer parameterization); MO seemed like a better place to look for one, since the issue of the composition of binary quadratic forms is notoriously challenging.

Is a complete solution or parameterization already known?

If not, what's the right way to go about deriving one? I can imagine trying to resolve $x^2+ay^2=mn^2$, where [hopefully] $m$ is squarefree; doing the same for $u^2+bv^2=jk^2$ would then leave $p^2+cq^2=jm(kn)^2 = j'm'(\delta kn)^2$ where $\delta=\gcd(j,m)$, which could be resolved the same way. Is that likely the best approach?

The main application is the solution of higher-degree equations which have been reduced to the form ($\star$).

cf.Barnett & Mendel, or Bradley). This would be a special case of that, I would imagine. $\endgroup$2more comments