Let K be a quadratic extension
of Q, B a quaternion algebra over Q and A = B ⊗QK. Let 𝒪 be a maximal
order in A extending an order in B. The projective norm one group P𝒪1 is
shown to be isomorphic to the spinorial kernel group O′(L), for an explicitly
determined quadratic Z-lattice L of rank four, in several general situations. In
other cases, only the local structures of 𝒪 and L are given at each prime.
Both definite and indefinite lattices are covered. Some results for quadratic
global field extensions K∕F and maximal S-orders are given. There is a
description of the F-quaternion subalgebras of A, and also of their norm one
groups as stabilizer subgroups and as unitary groups. Conjugacy classes of
the Fuchsian subgroups of P𝒪1 corresponding to stabilizer subgroups are
studied.
