In two famous papers [16], [17], Fong and Seitz showed that all finite Moufang generalized polygons were classical or dual classical. In fact, they obtained this result in group theoretical terms (classifying finite split BN-pairs), but Tits remarked the simple geometrical translation. And of course, the converse was already well known. In a search for a synthetic “elementary” proof of the Fong–Seitz result for the specific case of generalized quadrangles (which is the central and most difficult part in [16], [17]), Payne and . Thas noticed that when one looks at the group generated by all rootelations and dual root-elations which stabilize a given point of a.
