Quaternion order (awaiting review)

Let $B$ be a quaternion algebra over $\Q$. An order $O\subset B$ is a $\Z$-lattice in $B$ that is also a subring of $B$ (where we require $1 \in O$). We say that $O$ is maximal if it is not properly contained in another order of $B$.

Unlike orders in number fields, there is never a unique maximal order in a rational quaternion algebra; however, the number of $B^\times$-conjugacy classes is finite (and in many cases, there is exactly one $B^\times$-conjugacy class).