Quadratic diophantine equations and fundamental solutions I posted a question about a class of ternary quadratic equations. However, after some searching, I found the problem is not so simple as I thought. Thus, I want to know just whether a property on the Pell equations can be generalized.
Let $a $ and $b $ be integers, $\ne 0, 1, -1$. Consider a quadranary diophantine equation $x^2-ay^2-bz^2+abw^2=1$. The "grand" problem is to classify "nontrivial" (that is, $x^2\ne 1$) solutions. The existence may follow from the classical theory of Pell's equations.
Then, I want to know whether there are always(or for almost all cases) a finite set of "fundamental" solutions, generating the other solutions by some method. In case of Pell's equations, there is a unique fundamental solution $x_1+\sqrt{n}y_1$ for $x^2-ny^2=1$ and the other solutions are given as $x_n+\sqrt{n}y_n = (x_1+\sqrt{n}y_1)^n$. To utilize a similar way, I chose the class of equations to lie in the unit group of a quaternion algebra $(a,b)_\mathbb {Q} $.

I guess Majid Jahangiri's article, which I found a moment ago: >>Generators of arithmetic quaternion groups and a diophantine problem<< is a quite useful starting point.
Jahangiri uses the notion of the "Ford fundamental domain", which is entirely new for me. I need to study more geometric aspects of these equations. After then, I expect to be able to pose a more interesting question.
