Optimal spinor selectivity for quaternion orders

Let $D$ be a quaternion algebra over a number field $F$, and $\mathscr{G}$ be an arbitrary genus of $O_F$-orders of full rank in $D$. Let $K$ be a quadratic field extension of $F$ that embeds into $D$, and $B$ be an $O_F$-order in $K$ that can be optimally embedded into some member of $\mathscr{G}$. We provide a necessary and sufficient condition for $B$ to be optimally spinor selective for the genus $\mathscr{G}$, which generalizes previous existing optimal selectivity criterions for Eichler orders as given by Arenas, Arenas-Carmona and Contreras, and by Voight independently. This allows us to obtain a refinement of the classical trace formula for optimal embeddings, which will be called the spinor trace formula. When $\mathscr{G}$ is a genus of Eichler orders, we extend Maclachlan's relative conductor formula for optimal selectivity from Eichler orders of square-free levels to all Eichler orders.