Remark 2

Let F be a finite-dimensional real affine space and G a group of automorphisms of F. For all $g \in G$, denote by U(g) the automorphism of the space of translations V of F associated to g. Assume that the image U(G) is a finite subgroup of $\mathbf{GL}(V)$; then V has a scalar product invariant under U(G) (Integration, Chap. VII, § 3, no. 1, Prop. 1). If, in addition, G acts properly on F when it is provided with the discrete topology, and if it is generated by reflections, we can apply to G the results of this paragraph.