Galois involutions and exceptional buildings (Q515313)

The theory of descent in buildings developed in [\textit{B. Mühlherr} et al., Descent in buildings. Princeton, NJ: Princeton University Press (2015; Zbl 1338.51002)] deals with the connections between a spherical building and its ambient split building. The authors now apply this theory to give elementary constructions of the exceptional buildings that arise as fixed-point buildings of Galois involutions. The first 9 sections provide background material. The later sections construct the Moufang quadrangles of type \(\mathrm{E}_8\), \(\mathrm{E}_7\), \(\mathrm{E}_6\), the exceptional buildings of type \(\mathrm{A}_2\) (Moufang planes obtained from octonion division algebras) and the non-pseudo-split buildings of type \(\mathrm{F}_4\) (obtained from certain composition algebras). The last two sections construct the Moufang quadrangles of type \(\mathrm{F}_4\), in a similar fashion as in [\textit{B. Mühlherr} and \textit{H. Van Maldeghem}, Can. J. Math. 51, No. 2, 347--371 (1999; Zbl 0942.51002)].