Transfer to the root-action Brink–Howlett groupoid

The object set is the same as that of $\mathcal{B}\mathcal{H}(J)$ in [F1]. Its arrow condition $x{\mathrm{\Pi }}_K={\mathrm{\Pi }}_I$ is exactly the condition for $(I,x,K)$ in $\mathcal{B}$. For article arrows $x:I\to K$ and $y:K\to L$, the corresponding composite in $\mathcal{B}$ is $(I,x,K)(K,y,L)=(I,xy,L)$. Thus reversal preserves the label of the article composite, as asserted.

Let $E$ be any component for which $W_E$ is finite. Distinct indices give distinct simple generators: the generator $s_i$ sends ${\alpha }_i$ to $-{\alpha }_i$, while for $j\mathrm{\ne }i$ the ${\alpha }_i$ coefficient of $s_j{\alpha }_i={\alpha }_i-2B({\alpha }_i,{\alpha }_j){\alpha }_j$ is $1$. Hence $E$ is finite. By [F2], the restricted system on $E$ identifies with $W_E$, with lengths preserved. The ambient reflection formulas restrict on the span of its simple roots to its canonical representation. Applying [F3] to this finite system therefore gives $w_E^2=1$ and $w_E{s}_i{w}_E=s_{{\iota }_E(i)}$ with the same opposition permutation used for the article label. Applying the root formula twice gives ${\alpha }_i={\alpha }_{{\iota }_E^2(i)}$; the distinct basis vectors imply ${\iota }_E^2=1$.

For $a\in E$ put $b={\iota }_E(a)$, so ${\iota }_E(b)=a$. Conjugation by $w_E$ permutes the simple generators of $W_E$ and sends the generators indexed by $E\setminus \{b\}$ bijectively to those indexed by $E\setminus \{a\}$. It preserves length: replacing each letter in a shortest word gives one inequality, and the inverse conjugation gives the reverse inequality. Thus it carries the maximum-length element of $W_{E\setminus \{b\}}$ to the maximum-length element of $W_{E\setminus \{a\}}$. Uniqueness in [F3] gives $w_E{w}_{E\setminus \{b\}}{w}_E=w_{E\setminus \{a\}}$.

For an article elementary arrow put $E=[I,a]$, $b={\iota }_E(a)$ and $K=(I\cup \{a\})\setminus \{b\}$. Its label is $v_{a,I}=w_{E\setminus \{a\}}{w}_E$. Step 2.1 and $w_E^2=1$ give $v_{a,I}=w_E{w}_{E\setminus \{b\}}=\nu (b,K)$ by the imported single-root formula. Indeed $K\cup \{b\}=I\cup \{a\}$, so the component containing $b$ is precisely $E$. By step 1.1 this generator has source $K$ and target $I$ in $\mathcal{B}$. Both are associates of $J$ by [F1], so it belongs to the restricted groupoid. Only $W_E$ must be finite.

Conversely, let $g=\nu (b,K)=w_E{w}_{E\setminus \{b\}}$ label a Brink–Howlett generator of the restricted groupoid, where $E=[K,b]$ is spherical. The cited inverse-generator relation says that its inverse is again a Brink–Howlett generator. Set $a={\iota }_E(b)$ and $I=(K\cup \{b\})\setminus \{a\}$. By the finite-parabolic involutions in [F3], $g^{-1}=w_{E\setminus \{b\}}{w}_E=v_{b,K}$. By [F1] this is the article elementary arrow $K\to I$, so $I$ is an associate and $g{\mathrm{\Pi }}_K={\mathrm{\Pi }}_I$. Since $I\cup \{a\}=K\cup \{b\}$, one has $[I,a]=E$; step 1.2 gives ${\iota }_E(a)=b$. Step 2.1 now yields $v_{a,I}=w_{E\setminus \{a\}}{w}_E=w_E{w}_{E\setminus \{b\}}=g$. Thus $g$ is exactly the reversal of the article elementary arrow $I\to K$, proving the converse without any finiteness condition on the other components.