It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex plane: $\mathbb{Q}[i]$ and $\mathbb{Q}[e^{2\pi i/3}]$. Can I generalize this fact to the quaternionic(?) extension of $\mathbb{Q}$? That is,
how many (up to conjugacy?) division algebras $D$ over $\mathbb{Q}$ are there, with:
(i) $D\subset \mathbb{H}$, where $\mathbb{H}$ is the quaternion skew-field,
(ii) $D$ is a 4-dimensional vector space over $\mathbb{Q}$ containing $\mathbb{Q}$,
(iii) for some unital(norm 1) base $B=\{1,a,b,c\}$ of $D$, the span $B\mathbb{Z}$ forms a 4-dim lattice,
(iv) $B$ in (iii) satisfies that for any $n\in\mathbb{N}$, $a^n,b^n,c^n\in B\mathbb{Z}$?
I cannot be completely convinced that the problem is well-posed, but I guess the point seems clear.