  Let $(W,S)$ be a Coxeter system and $\Gamma$ be a group of automorphisms of
$W$ such that $\gamma(S)=S$ for all $\gamma \in \Gamma$. Then it is known that
the group of fixed points $W^\Gamma$ is again a Coxeter group with a
canonically defined set of generators. The usual proofs of this fact rely on
the reflection representation of $W$. Here, we give a proof which only uses the
combinatorics of reduced expressions in $W$. As a by-product, this shows that
the length function on $W$ restricts to a weight function on $W^\Gamma$.
