Moufang sets related to polarities in exceptional Moufang quadrangles of type $\mathsf{F}_4$

Abstract

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type ${\mathsf{F}}_4$, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Moufang quadrangle of type ${\mathsf{F}}_4$ stabilizing the absolute points of a polarity, also centralizes the polarity. This handles the final case of a similar result for all polarities of Moufang $n$-gons with $n\ge 3$.