Optimal spinor selectivity for quaternion Bass orders

Let \(A\) be a quaternion algebra over a number field \(F\), and \(\mathcal{O}\) be an \(O_F\)-order of full rank in \(A\). Let \(K\) be a quadratic field extension of \(F\) that embeds into \(A\), and \(B\) be an \(O_F\)-order in \(K\). Suppose that \(\mathcal{O}\) is a Bass order that is well-behaved at all the dyadic primes of \(F\). We provide a necessary and sufficient condition for \(B\) to be optimally spinor selective for the genus of \(\mathcal{O}\). This partially generalizes previous results on optimal (spinor) selectivity by C. Maclachlan [Optimal embeddings in quaternion algebras. J. Number Theory, 128(10):2852-2860, 2008] for Eichler orders of square-free levels, and independently by M. Arenas et al. [On optimal embeddings and trees. J. Number Theory, 193:91-117, 2018] and by J. Voight [Chapter 31, Quaternion algebras, volume 288 of Graduate Texts in Mathematics. Springer-Verlag, 2021] for Eichler orders of arbitrary levels.