The exceptional Tits quadrangles revisited (Q6655963)

A Tits polygon is a bipartite graph in which the neighborhood of each vertex is endowed with an opposition relation satisfying certain axioms which reduce to the axioms of a Moufang polygon in the case that the opposition relations are all trivial. There is a standard construction (see the authors [\textit{B. Mühlherr} et al., Tits polygons. With an appendix by Holger P. Petersson. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1498.51001)]) that produces a Tits polygon from a pair \((\Delta, T)\), where \(\Delta\) is an arbitrary irreducible spherical building of rank at least \(3\) and \(T\) is a suitable Tits index. In an earlier paper [Transform. Groups 25, No. 4, 1289--1344 (2020; Zbl 1477.51004)], the authors have characterized the exceptional Tits quadrangles as extensions of orthogonal Tits quadrangles.\N\NIn the article under review they complete the proof of a characterization of the Tits quadrangles that arise in this way from the spherical building associated to an exceptional algebraic group.