Integer solutions of Quaterni0n norm

Question

I would like to know all integer solutions of the Diophantine equation $x^2-ay^2-bz^2+abw^2=1$ where $a,b$ are fixed positive integers. If you know the answer, I appreciate your reply. (To be more specific, $a,b$ are such that $x^2-ay^2-bz^2+abw^2=0$ has no solution)

Answer 1

The simplest approach I think is.

$$x^2-ay^2-bz^2+abw^2=1$$

To make the change. $y=kS$ ; $z=tS$ ; $w=pS$

Reduced to a Pell equation. That factor was not square. And using his solution we write the answer.

$$x^2-(ak^2+bt^2-abp^2)S^2=1$$