The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\alpha$ be a totally real algebraic integer, and $F=\mathbb{Q}(\alpha)$. Consider the quadratic equation $$4x^2+4y^2+(4-\alpha^2)(z^2+w^2)=4.$$ It is known by quaternions that the set of solutions of the equation forms an arithmetic group in products of SL_2($\mathbb{R}$), hence definitely nontrivial. However, it is hard to write down particular ones besides $(\pm 1,0,0,0)$ and $(0,\pm 1,0,0)$. In general, consider the Chebyshev polynomials of the first (second) kind $T_n$ ($U_n$). If $T_n(\frac{\alpha}{2})$ is an algebraic integer, then $(\pm T_n(\frac{\alpha}{2}), 0, \pm U_{n-1}(\frac{\alpha}{2}),0)$ or switch the last two coordinates are solutions, too.
My questions are, can we find other solutions precisely? Or consider the solutions with $y^2+(4-\alpha^2)w^2\neq 0,1$, can we find a lower bound for $y^2+(4-\alpha^2)w^2$ in terms of a function of $|\alpha|$? The above example has a bound related to the degree of $\alpha$, hence not the absolute value of $\alpha$.
I am not very familiar with this direction. Any good references for the solutions of quadratic equations over number rings will be helpful.
