Lipschitz monoids and Vahlen matrices

Recent studies of Vahlen matrices have again pointed out a very interesting multiplicative monoid that is present in every Clifford algebra; this motivates a new presentation of the knowledge already collected about this monoid before it got involved in Vahlen matrices. This monoid is first studied for itself (and under rather weak hypotheses), and secondly in view of a more effective application to Vahlen matrices. The “invariance property” of this monoid opens the way to new information about these matrices.