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} $.

Suggest beginning with Carmichael, manuscript pages 30-35, pdf pages 42-47.  
For any integer quadruple $(p,q,r,s)$ such that $p^2 - a q^2 - b r^2 + ab s^2 \neq 0,$ we get a rational solution 
$$ (1 + tp, \; tq, \; tr, \; ts)  $$ where
$$ t = \frac{-2p}{p^2 - a q^2 - b r^2 + ab s^2}.  $$
This does include all integer solutions, but selecting those is likely to be a mess. The main problem is that you take a fixed right hand side, $1,$ rather than $w^2$ with another variable, so that the whole thing is the null set of an indefinite quadratic form. In this latter problem you could clear denominators by taking $w = p^2 - a q^2 - b r^2 + ab s^2,$ then divide out by the GCD of all five variables, giving primitive solutions. In some problems the set of possible GCD's to worry about is finite, once $\gcd(p,q,r,s) = 1.$ I have no idea whether such good fortune happens here. 
From the language of integer (indefinite) lattices, we have an indefinite dot product, the bilinear form given by the evident 4 by 4 matrix. Over the rationals, a fixed vector $v$ with nonzero norm gives a reflection
$$  x \mapsto x - \frac{2 \, x \cdot v}{v \cdot v} v.  $$
This is what is called an "odd lattice," so this preserves all integer points only when $v \cdot v = \pm 1.$ i think it likely that the full automorphism group of your indefinite form is generated by reflections. This is somewhat circular, since you need to find vectors of norm $\pm 1$ to find reflections, and that is what you wanted in the first place. However, the thing sort of mushrooms; given a few solutions, you get a few reflections, these act on the solutions you have, and so on. 
For quadratic forms, I recommend CASSELS  since it emphasizes $\mathbb Q$ and $\mathbb Z.$ Note that MAJID JAHANGIRI alludes to the book Fricke and Klein (1897) but that is not in the references. A fair amount of the relevant material is availbale, in English in MAGNUS.
This is nice, you can read several pages of CHALK 
