Yet another way to express this can be found in Chapter IV, comment after Lemma 1.0 page 180 of arxiv.org/pdf/1605.04832.pdf for algebraic structures defined by generators and relations.

Suslin's paper is "Quaternion homomorphism for the field of functions on a conic" and can be found online. Only beginning of page 2 is relevant here. It refers to a result of Dennis presented in the paper "On K 2 of finite dimensional division algebras over arithmetical fields" by Rehmann and Stuhler. By "unfolding" I meant something like you indicate.