Arithmetic Kleinian group

A Kleinian group $\Gamma$ is arithmetic if it is commensurable up to conjugacy in $\operatorname{PSL}_2(\mathbb C)$ with the projective image of $\rho(\mathcal O^1)$, for data satisfying all of the following:

• $K$ is a number field with exactly one conjugate pair of nonreal complex embeddings;
• $B$ is a quaternion algebra over $K$, ramified at every real embedding;
• $\mathcal O\subset B$ is an order and $\mathcal O^1$ is its norm-one group;
• $\rho:B\hookrightarrow M_2(\mathbb C)$ comes from a splitting $B\otimes_{K,\sigma}\mathbb C\cong M_2(\mathbb C)$ at one of the nonreal embeddings $\sigma$.

“One complex place” means one conjugate pair, not one individual embedding. The finite-index commensurability requirement is essential; an arbitrary infinite-index subgroup of such a group need not be arithmetic in this sense.