Supplement on Curved flats in the space of point pairs and Isothermic surfaces: A Quaternionic Calculus

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvature in certain space forms is presented --- in particular, a new form of Bryant's Weierstrass type representation for surfaces of constant mean curvature 1 in hyperbolic 3-space is given.