Algebraic structures related to universal covers of special rank one groups

In this note we give a presentation of special rank one groups with abelian unipotent subgroups (abbreviated AUS), depending on some algebraic structure K, and then show that each special rank one group with AUS is a center-factor-group of a so presented group. As shown in [T. De Medts, R. Weiss, Moufang sets and Jordan division algebras, Math. Ann. 335 (2006) 415–433] one example of such an algebraic structure is unital quadratic Jordan Division Algebras. In subsequent work my student Sebastian Weiß studied these structures, called R-structures, systematically and showed, among other things, that the problem of classifying such special rank one groups with AUS is equivalent to the classification of these R-structures up to Isotopy.