Lemma 2

The orthogonal complement $E^0$ of $E$ for $B_M$ is of dimension 1; it is generated by an element $v = \sum_{s \in S} v_s e_s$ with $v_s > 0$ for all $s$.

This follows from Lemma 4 of $§ 3$, no. 5, applied to the matrix with entries $B_M(e_s, e_{s'})$.

Let $v = \sum_s v_s e_s$ be the vector satisfying the conditions of Lemma 2 and such that $\sum_s v_s = 1$, and let $A$ be the affine hyperplane of $E^*$ consisting of the $y^* \in E^*$ such that $\langle v, y^* \rangle = 1$. If $T$ denotes the orthogonal complement of $v$ in $E^*$, $A$ has a natural structure of an affine space with space of translations $T$. Moreover, the form $B_M$ defines by passage to the quotient a non-degenerate scalar product on $E/E^0$, hence also on its dual $T$; this gives a euclidean structure on the affine space $A$ (Algebra, Chap. IX, § 6, no. 6).

Let $G$ be the subgroup of $\mathbf{GL}(E)$ consisting of the automorphisms leaving $v$ and $B_M$ invariant; if $g \in G$, the contragredient map ${}^t g^{-1}$ leaves $A$ and $T$ stable, and defines by restriction to $A$ a displacement $i(g)$ of $A$ (cf. § 3). It is immediate that this gives an isomorphism from $G$ to the group of displacements of $A$. Moreover, the stabiliser $G_a$ of a point $a$ of $A$ can be identified with the orthogonal group of the Hilbert space T and is thus compact. On the other hand, G is a locally compact group countable at infinity and A is a Baire space: it follows (Integration, Chap. VII, Appendix, Lemma 2) that the map $\psi : g \mapsto g(a)$ defines a homeomorphism from $G/G_a$ to A. Thus G acts properly on A (General Topology, Chap. III, § 4, no. 2, Prop. 5). Since W is a subgroup of G, it can be identified with a group of displacements of A. We are going to show that this group satisfies the assumptions of § 3. More precisely: