Spectral selections, commutativity preservation and Coxeter-Lipschitz maps

Let $(W,S)$ be a Coxeter system whose graph is connected, with no infinite edges. A self-map $\tau$ of $W$ such that ${\tau }_{\sigma \theta }\in \{{\tau }_{\theta },\sigma {\tau }_{\theta }\}$ for all $\theta \in W$ and all reflections $\sigma$ (analogous to being 1-Lipschitz with respect to the Bruhat order on $W$) is either constant or a right translation. A somewhat stronger version holds for $S_n$, where it suffices that $\sigma$ range over smaller, $\theta$-dependent sets of reflections. These combinatorial results have a number of consequences concerning continuous spectrum- and commutativity-preserving maps $\mathrm{S}\mathrm{U}(n)\to M_n$ defined on special unitary groups: every such map is a conjugation composed with (a) the identity; (b) transposition, or © a continuous diagonal spectrum selection. This parallels and recovers Petek’s analogous statement for self-maps of the space $H_n\le M_n$ of self-adjoint matrices, strengthening it slightly by expanding the codomain to $M_n$.