Abstract
This chapter considers the action of a strictly semi-linear automorphism fixing a root on the corresponding root group. It begins with the hypothesis whereby Δ is a Moufang spherical building and Π is the Coxeter diagram of Δ; here the chapter fixes an apartment Σ of Δ and a root α of Σ. The discussion then turns to a number of assumptions about an isomorphism of Moufang sets, anisotropic quadratic space, and root group sequence, followed by a lemma where E is an octonion division algebra with center F and norm N and D is a quaternion subalgebra of E. The chapter concludes with three versions of what is really one result about fixed points of non-linear automorphisms of the Moufang sets.
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