Let $B$ be a definite quaternion algebra and $O$ a maximal order of $B$. In proposition 5.3.1. of Vigneras' "The arithmetic of quaternion algebra", the author classifies $O^*$. But there's no proof. How can we show it?
I think that the point is theorem 1.3.7. in this book, which classifies finite groups of the division quaternion over the real numbers.
I know that for $ u \in O^*$, the order of $u$ is $2,4$ or $6$. Hence by theorem 1.3.7., $O^*$ is isomorphic to the cyclic group of order $2,4,6$, the dihedral group of order $4,8,12$, the "bilinear tetrahedral group", the "bilinear octahedral group", or the "bilinear icosahedral group".
How can we exclude the later 2 cases? And how can we show that if the discriminant of $B/\mathbb{Q}$ is neither $2$ nor $3$, then $O^*$ is cyclic? And how can we compute $O^*$ for the case that the discriminant is $2$ or $3$?