Polarized quaternion order (awaiting review)

Let $O$ be an order in an indefinite division quaternion algebra $B$ over $\Q$. Let $\operatorname{trd} : B \to \Q$ be the reduced trace. A polarization of $O$ is an element $\mu \in B^\times$ such that $\mu^2 \in \Z_{<0}$ and for all $x \in O$, we have $\operatorname{trd}(\mu x) \in \Z$.

A polarized order is a pair $(O,\mu)$ where $O$ is an order in an indefinite division quaternion algebra $B$ over $\Q$ and $\mu \in B^\times$ is a polarization of $O$.