A word property for twisted involutions in Coxeter groups

Given an involutive automorphism θ of a Coxeter system (W, S), let I(θ)⊆W denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced S_-expressions for any given w∈I(θ). This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.