Remark

We retain the notation of Prop. 5 and assume in addition that $K = \mathbf{R}$. Put $s = s_H$ and $s' = s_{H'}$. Let $H_n$ be the hyperplane $H + n.v$ and let $C_n$ be the set of points of $E$ of the form $a + \xi.v$ with $a \in H$ and $n < \xi < n + 1$. The $C_n$ are connected open sets forming a partition of $E - \bigcup_n H_n$. They are therefore the chambers defined by the system $\mathfrak{H} = (H_n)_{n \in \mathbf{Z}}$ in $E$. The translation $(s's)^n$ transforms the chamber $C = C_0$ into the chamber $C_{2n}$, and since $s(C_0) = C_{-1}$, we have $(s's)^n s(C) = C_{2n-1}$. It follows that the dihedral group W generated by s and s' permutes the chambers Cn simply-transitively*. Moreover, as we shall now show, if the chambers C and w(C) are on opposite sides of H (for $w \in W$), we have l(sw) = l(w) - 1 (the lengths being taken with respect to $S = \{s, s'\}$ (Chap. IV, § 1, no. 1)). Indeed, we then have $w(C) = C_n$ for some $n < 0$. If $n = -2k$, then $w = (ss')^k$ and $sw = (s's)^{k-1}s'$, so $l(w) = 2k$ and $l(sw) = 2k - 1$ (Chap. IV, § 1, no. 2, Remark). If $n = -2k - 1$, then $w = (ss')^k s$ and $sw = (s's)^k$, so $l(w) = 2k + 1$ and $l(sw) = 2k$.