Reduced words, root signs and finite coroot inversions

By F2 and F3 all real roots are roots and have one sign. If a positive real root $\beta$ is not ${\alpha }_i$, F2 implies that some coefficient at ${\alpha }_j$ with $j\mathrm{\ne }i$ is positive. Reflection $s_i$ leaves that coefficient unchanged by F4; its image is a root by F3, so its one-sign property forces it to stay positive. Since $s_i^2=1$ and only ${\alpha }_i$ maps to $-{\alpha }_i$, this restriction is a permutation. Apply the identical statements F2 and F3 to the transpose realization of F1 to obtain both conclusions for coroots. Nonzero vectors with independent coordinates cannot have both signs.

We need lifts with the full Cartan action. For $D=\mathrm{ad}{e}_i$, F6 bounds powers on $e_j$; F5 gives $D{f}_j={\delta }_{ij}{h}_i$, $D^2{f}_i=-2e_i$, $D^3{f}_i=0$, and $D^{2}h=0$. The corresponding formulas and F6 bound $\mathrm{ad}{f}_i$ on all generators. The derivation identity $D^m[x,y]={\sum }_k\left(\genfrac{}{}{0px}{}{m}{k}\right)[D^{k}x,D^{m-k}y]$, obtained inductively from the Leibniz rule, propagates these bounds to finite bracket words and sums. Thus their exponentials are pointwise finite Lie automorphisms: that identity proves bracket preservation, and the inverse exponential follows from the finite binomial expansion of $\exp (D)\exp (-D)$. Set $T_i=\exp (\mathrm{ad}{f}_i)\exp (-\mathrm{ad}{e}_i)\exp (\mathrm{ad}{f}_i)$. The relations give successive images of $h_i$ equal to $h_i+2f_i$, $-h_i+2f_i$, and $-h_i$. Each exponential fixes $\ker {\alpha }_i\subset \mathfrak{h}$. Splitting $h=(h-{\alpha }_i(h)h_i\mathrm{/}2)+{\alpha }_i(h)h_i\mathrm{/}2$ therefore proves $T_{i}h=s_{i}h$. For $x\in {\mathfrak{g}}_{\beta }$, $[h,T_{i}x]=T_i[s_{i}h,x]=(s_i\beta )(h)T_{i}x$, so it has the required root action as well.

Suppose $v{\alpha }_i={\alpha }_j$ for a Weyl word $v$, and lift that word by the product of automorphisms in 1.2. It maps $[{\mathfrak{g}}_{{\alpha }_i},{\mathfrak{g}}_{-{\alpha }_i}]=\mathbb{C}{h}_i$ onto $[{\mathfrak{g}}_{{\alpha }_j},{\mathfrak{g}}_{-{\alpha }_j}]=\mathbb{C}{h}_j$ by F2 and F5. Since its Cartan action is $v$, write $v{h}_i=c{h}_j$, with $c\mathrm{\ne }0$. Duality gives $2={\alpha }_i(h_i)=(v{\alpha }_i)(v{h}_i)=2c$, hence $c=1$. F4 now gives $v{s}_i{v}^{-1}(\lambda )=\lambda -\lambda (v{h}_i)v{\alpha }_i=s_j\lambda$ on the entire dual Cartan.

Suppose $w=s_{i_1}\cdots s_{i_t}$ sends ${\alpha }_i$ to a negative root. Track suffix images from ${\alpha }_i>0$ at the right to $w{\alpha }_i<0$ at the left. At a positive-to-negative transition at position $r$, put $v=s_{i_{r+1}}\cdots s_{i_t}$. By 1.1, $v{\alpha }_i={\alpha }_{i_r}$. Step 2.1 gives $v{s}_i=s_{i_r}v$, so $w{s}_i=s_{i_1}\cdots s_{i_{r-1}}{s}_{i_r}v{s}_i=s_{i_1}\cdots s_{i_{r-1}}v$, deleting the $r$th factor. This is valid even for a nonreduced input word and even for an empty suffix.

Apply 3.1 to a minimal word for $w$. If $w{\alpha }_i<0$, it gives $\mathrm{\ell }(w{s}_i)\le \mathrm{\ell }(w)-1$. If $w{\alpha }_i>0$, then $(w{s}_i){\alpha }_i=-w{\alpha }_i<0$, so the same argument for $w{s}_i$ gives $\mathrm{\ell }(w)<\mathrm{\ell }(w{s}_i)$. These are the only signs by 1.1, proving the first iff. Repeat 1.2–3.1 for the transpose GCM, which has the same word lengths by F1, to get $\mathrm{\ell }(w{s}_i)<\mathrm{\ell }(w)$ iff $w{h}_i<0$. Thus both equivalences hold without any Coxeter presentation.

For finiteness, let $w$ have finite inversion set and consider $w{s}_i$. On positive coroots other than $h_i$, the bijection $\beta \mapsto s_i\beta$ from 1.1 identifies their inversions for $w{s}_i$ with the inversions of $w$ other than a possible $h_i$. At $h_i$, $(w{s}_i)h_i=-w{h}_i$, so its inversion status is the opposite of its status for $w$. Therefore the cardinality changes by exactly $1$ or $-1$, and in particular by at most $1$. Starting with $\mathrm{Inv}(1)=\mathrm{\varnothing }$ and inducting along any word proves finiteness and the bound by that word's length; use a minimal word for the stated bound. The empty word and empty simple system give zero inversions; a single reflection has just $h_i$. No infinite choices, root bases or finiteness of $W$ enter.