Rotations and units in quaternion algebras (original) (raw)

Special orthogonal group of the space of pure quaternions in a quaternion algebra over a quadratic field Unit groups of orders in quaternion algebras over number fields provide important examples of non-commutative arithmetic groups. Let K = Q( √ d ) be a quadratic field with d < 0 a squarefree integer such that d ≡ 1(mod 8), and let R be its ring of integers. In this note we study, through its representation in SO 3 (R), the group of units of several orders in the quaternion algebra over K with basis {1, i, j, k} satisfying the relations i 2 = j 2 = -1, ij =ji = k.