The real Coxeter form, its radical, reflections, and form-preserving maps

Let $S$ be a finite set and let $m$ be a Coxeter matrix on $S$, so that $m(s,s)=1$ and $m(s,t)=m(t,s)\in \{2,3,\dots \}\cup \{\mathrm{\infty }\}$ for $s\mathrm{\ne }t$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

(1) The space and its distinguished functions. Let $V:={\mathbb{R}}^S$ be the real vector space of all functions $S\to \mathbb{R}$, with pointwise operations (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$); it is a vector space over the ordered field $\mathbb{R}$ (The reals form a totally ordered field). Write $u(s)$ for the value of $u$ at $s$ and put $e_s(s):=1$, $e_s(t):=0$ for $t\mathrm{\ne }s$.

(2) The Coxeter form. For finite $m(s,t)$ put $c(s,t):=\cos (\pi \mathrm{/}m(s,t))$ (Sine and cosine defined by their real power series, Pi as twice the smallest positive zero of cosine) and for $m(s,t)=\mathrm{\infty }$ put $c(s,t):=1$; then $c(s,t)=c(t,s)$ and $c(s,s)=\cos \pi =-1$. The Coxeter form $B$ is the unique symmetric bilinear form on $V$ with $B(e_s,e_t)=-c(s,t)$ for all $s,t\in S$; in particular $B(e_s,e_s)=1$ and $B(e_s,e_t)=-\cos (\pi \mathrm{/}m(s,t))$ for finite $m(s,t)$, while $B(e_s,e_t)=-1$ when $m(s,t)=\mathrm{\infty }$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V\to V^{\ast }$). No positive definiteness, definiteness or nondegeneracy of $B$ is presumed: $\mathrm{rad}(B):=\{u:B(u,v)=0\text{ for all }v\in V\}$ may be nonzero (The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space), and the form may be indefinite (Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form). A linear map $g:V\to V$ is $B$-preserving when $B(gu,gw)=B(u,w)$ for all $u,w\in V$.

(3) Reflections. For $a\in V$ with $B(a,a)\mathrm{\ne }0$ define the reflection with normal $a$ by $r_a(v):=v-\frac{2B(v,a)}{B(a,a)}a\mathrm{.}$ The definition asserts neither that $r_a$ is linear or an involution nor that $\ker B(-,a)=\{v:B(v,a)=0\}$ is a hyperplane; these are proved in Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order ↗. When $B(a,a)=0$ the symbol $r_a$ is not defined.