We classify all domestic collineations, that is, collineations mapping no chamber to an opposite one, of all thick spherical buildings of type \(\mathsf {F_4}\) . Besides the previously known cases like central elations and products of two perpendicular such elations, we find collineations that pointwise fix certain subspaces, also of type \(\mathsf {F_4}\) , but over a smaller algebra, or even non-thick as a building. We also find examples that pointwise fix Moufang quadrangles, and these inclusions are new: Moufang quadrangles of absolute type \(\mathsf {D_5}\) are contained in buildings of type \(\mathsf {F_4}\) of absolute type \(\mathsf {E_6}\) , and exceptional Moufang quadrangles of type \(\mathsf {E_6}\) are found inside buildings of relative type \(\mathsf {F_4}\) and absolute type \(\mathsf {E_7}\) (the so-called quaternion metasymplectic spaces). Together with the already established Moufang quadrangles of mixed type inside mixed buildings of type \(\mathsf {F_4}\) , our results imply that domestic collineations give rise to inclusions of the three different types of Moufang quadrangles inside metasymplectic spaces: Moufang quadrangles of classical, exceptional and mixed type.
