{J}} {\mathbb{Z}}_{\ge 0} \alpha_i^{\vee}$, ${\Delta_{J}}:= \Delta \cap {Q_{J}}$, ${\Delta_{J}}^{+} := \Delta^{+} \cap {Q_{J}}$, and ${W_{J}}:= \langle s_{i} \mid i \in {J}\rangle$. Also, we denote by $$\label{eq:prj}
[\,\cdot\,]_{{J}} : 
Q^{\vee} \twoheadrightarrow {Q_{J}^{\vee}}\quad 
\text{(resp., $[\,\cdot\,]^{{J}} : Q^{\vee} \twoheadrightarrow Q_{I \setminus {J}}^{\vee}$)}$$ the projection from $Q^{\vee}=Q_{I \setminus {J}}^{\vee} \oplus {Q_{J}^{\vee}}$ onto ${Q_{J}^{\vee}}$ (resp., $Q_{I \setminus {J}}^{\vee}$) with kernel $Q_{I \setminus {J}}^{\vee}$ (resp., ${Q_{J}^{\vee}}$). Let ${W^{J}}$ denote the set of minimal(-length) coset representatives for the cosets in $W/{W_{J}}$; we know from [@BB Sect. 2.4] that $$\label{eq:mcr}
{W^{J}}= \bigl\{ w \in W \mid 
\text{$w \alpha \in \Delta^{+}$ for all $\alpha \in {\Delta_{J}}^{+}$}\bigr\}.$$ For $w \in W$, we denote by ${\lfloor w \rfloor}={\lfloor w \rfloor}^{{J}} \in {W^{J}}$ the minimal coset representative for the coset $w {W_{J}}$ in $W/{W_{J}}$. Also, following [@Pet97] (see also [@LS10 Sect. 10]), we set $$\begin{aligned}
({\Delta_{J}})_{{\mathrm{af}}} 
  & := \bigl\{ \alpha + n \delta \mid 
  \alpha \in {\Delta_{J}},\,n \in {\mathbb{Z}}\bigr\} \subset \Delta_{{\mathrm{af}}}, \\
({\Delta_{J}})_{{\mathrm{af}}}^{+}
  &:= ({\Delta_{J}})_{{\mathrm{af}}} \cap {\Delta_{{\mathrm{af}}}^{+}}= 
  {\Delta_{J}}^{+} \sqcup \bigl\{ \alpha + n \delta \mid 
  \alpha \in {\Delta_{J}},\, n \in {\mathbb{Z}}_{> 0} \bigr\}, \\
\label{eq:stabilizer}
({W_{J}})_{{\mathrm{af}}} 
 & := {W_{J}}\ltimes \bigl\{ t_{\xi} \mid \xi \in {Q_{J}^{\vee}}\bigr\}
   = \bigl\langle s_{\beta} \mid \beta \in ({\Delta_{J}})_{{\mathrm{af}}}^{+} \bigr\rangle, \\
\label{eq:Pet}
({W^{J}})_{{\mathrm{af}}}
 &:= \bigl\{ x \in W_{{\mathrm{af}}} \mid 
 