  Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we
consider the subgroup of $W$ generated by the reflections in $W$ having
(-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we
determine completely when this subgroup is finitely generated, by using
preceding results on centralizers of reflections. This result provides a new
and naturally constructed example of a family of non-finitely generated
subgroups of finitely generated groups.
