Chamber walls of an essential finite reflection group give full Coxeter rank

By [F1] the group and its full hyperplane complement satisfy Davis's setup. The assumed equality $E^W=0$ supplies precisely its additional essentiality hypothesis. Thus [F2] makes $(W,S)$ a Coxeter system in the sense of Coxeter-system, roots, length and weak-order conventions, proves simple transitivity on chambers and makes $\bar{C}$ a full-dimensional simplicial cone with exactly $\dim E$ walls.

Put $n=\dim E$. By step 1.1, the full-dimensional simplicial cone $\bar{C}$ has the form $\{{\sum }_{i=1}^n{t}_i{v}_i:t_i\ge 0\}$ for a basis $v_1,\dots ,v_n$; its wall hyperplanes are $H_i=\ker {\phi }_i$, where ${\phi }_i$ is the coordinate functional from [F3]. The facet on $H_i$ is the cone generated by the $v_j$ with $j\mathrm{\ne }i$. By [F3], these ${\phi }_i$ form a basis of $E^{\ast }$.

The space $E$ is finite-dimensional and real, with the inner product of [F1], so [F4] applies. Put $u_i=J^{-1}({\phi }_i)$. Since $J$ is a linear bijection and the ${\phi }_i$ form a basis, the $u_i$ form a basis of $E$: applying $J$ proves independence and pulls back every linear combination in $E^{\ast }$. The equality $⟨x,u_i⟩={\phi }_i(x)$ makes $u_i$ a nonzero normal to $H_i$. Any other nonzero normal $z$ to $H_i$ has $J(z)(v_j)=0$ for $j\mathrm{\ne }i$, hence $J(z)=J(z)(v_i){\phi }_i$ by [F3]; injectivity gives $z=J(z)(v_i)u_i$, with nonzero scalar. Thus any choice of one nonzero wall normal per wall is a basis. There is one distinct reflection per wall by [F1], so $\mathrm{\mid }S\mathrm{\mid }=n=\dim E$ and all conclusions follow.