Bruhat order on the fixed-point subgroup by a Coxeter graph automorphism

Let (W,S) be a Coxeter system and σ be an automorphism of the Coxeter graph of (W,S). Then σ extends to a group automorphism of W and gives rise to a subgroup Wσ in W consisting of elements fixed by this automorphism σ. In this paper, we prove that Wσ is a Coxeter group, and that the Bruhat order on Wσ is the restriction of that on W.