Suppose $G$ is a subgroup of $O_n(\mathbb{R})$ generated by finitely many reflections, where a reflection is defined to be a linear transformation fixing a hyperplane and sending a normal vector to the hyperplane to its negative. Suppose the normal vectors corresponding to these hyperplanes are linearly independent. Is G necessarily isomorphic to a Coxeter group?
This problem arose upon considering the fact that every Coxeter group is isomorphic to its Tits reflection representation, and I was wondering if there was a converse statement. Looking at the literature, a common statement is that this is true for finite reflection groups, but I could not find a statement for finitely generated reflection groups.
I might be missing some easy counterexample here as it seems like it would be a standard result if it were true.
Thanks!