Question re parametric identity

Question

I am asking whether the existence of a parametric identity involving a set of quadruple signed integers $\{i, j, k, l\}$, which is true for infinitely many sets of quadruples for each integer value of $n, (n>0)$, implies that there is a relationship between some or all of the variables in the set $\{i, j, k, l\}$ and/or relationship between some (or all) of the parameters of the $\{i, j, k, l\}$ with $n$? And furthermore, I am asking whether such relationship would allow rewriting the parametric identity to hold true for subsets of the $\{i, j, k, l\}$ set that contain fewer than four parameters and perhaps involve direct relationship with $n$?

Answer 1

I wonder whether this is the kind of thing you have in mind:
$n=(i^2-j^2)-(k^2-l^2)$ has infinitely many integer solutions for each integer $n>0$.
I don't think this implies any relation among some or all of $i,j,k,l$ (unless you allow relations that also involve $n$).
But one does have $2r=(r+s+1)^2-(r+s)^2-((s+1)^2-s^2)$ which uses only two parameters, and has infinitely many solutions for each $r$.