Coincidence lattices in the hyperbolic plane

The problem of coincidences of lattices in the space R(p,q), with p + q = 2, is analyzed using Clifford algebra. We show that, as in R(n), any coincidence isometry can be decomposed as a product of at most two reflections by vectors of the lattice. Bases and coincidence indices are constructed explicitly for several interesting lattices. Our procedure is metric-independent and, in particular, the hyperbolic plane is obtained when p = q = 1. Additionally, we provide a proof of the Cartan-Dieudonné theorem for R(p,q), with p + q = 2, that includes an algorithm to decompose an orthogonal transformation into a product of reflections.