Vahlen groups defined over commutative rings

Elements of a Vahlen group are 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \times 2$$\end{document} matrices with entries in a Clifford algebra satisfying some conditions. They play a central role in the theory of higher dimensional harmonic automorphic forms. Traditionally they have come in both ordinary and paravector type and have been defined (over Clifford algebras) over the real or complex numbers. We extend the definition of both types to be over a commutative ring with an arbitrary quadratic form. We show that they are indeed groups and identify in each case the group as the pin group, spin group, or another subgroup of the Clifford group. Under some mild conditions, for both types we show the equivalence of our definition with a suitably generalised version of the two standard definitions.