's construction). The main point is that as we impose the restrictions one by one, at each stage, there should be a choice of $B$ and $R$ left. Most of the restrictions in (i)-(viii) are about ramification of $B$ at various places. Now a quaternion algebra with specified ramifications exists if and only if the number of places where it is ramified is even, which is condition (vii). We impose (i), and since (vii) can be satisfied while (i) holds, we have quaternion algebras $B$ satisfying (i) 