Automorphisms and opposition in spherical buildings of exceptional type, IV: The \(E_7\) case

An automorphism of a spherical building is called \textit{domestic} if it maps no chamber onto an opposite chamber. This paper forms a significant part of a large project classifying domestic automorphisms of spherical buildings of exceptional type. In previous work the classifications for \(\mathsf{G}_2\), \(\mathsf{F}_4\) and \(\mathsf{E}_6\) have been completed, and the present work provides the classification for buildings of type \(\mathsf{E}_7\). In many respects this case is the richest amongst all exceptional types.