an additional assumption, the so-called [*numerical condition*]{} which appeared in [@MM1 Formula (10)]. This additional assumption is rendered superfluous by [@MM6 Proposition 1].

Consequently,

**in what follows, we assume that $W$ is not simply laced.**

In type $G_2$, we have the following. $$\xymatrix{1\ar@{-}[r]^{6}&2}\qquad \qquad 
\begin{array}{c||c|c}
&\mathcal{L}_1&\mathcal{L}_2\\
\hline\hline
\mathcal{R}_1&1,121,12121&12,1212\\
\hline
\mathcal{R}_2&21,2121&2,212,21212
\end{array}$$

The general type $B_n$, for $n\geq 2$, corresponds to the Coxeter diagram $$\xymatrix{1\ar@{-}[r]^{4}&2\ar@{-}[r]&3\ar@{-}[r]&\dots\ar@{-}[r]&n}$$ and is given below: $$\begin{array}{c||c|c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\dots&\mathcal{L}_n\\
\hline\hline
\mathcal{R}_1&1,121&12&123&\dots&12\cdots n \\
\hline
\mathcal{R}_2&21&2,212&23, 2123&\dots&23\cdots n,212\cdots n \\
\hline
\mathcal{R}_3&321&32,3212&3,32123&\dots&34\cdots n,3212\cdots n \\
\hline
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots \\
\hline
\mathcal{R}_n&n\cdots 21&n\cdots 32,n\cdots212&n\cdots43,n\cdots 2123&\dots&n,n\cdots212\cdots n 
\end{array}$$

The remaining Weyl type $F_4$ looks as follows: $$\xymatrix{1\ar@{-}[r]&2\ar@{-}[r]^{4}&3\ar@{-}[r]&4}\qquad \qquad 
\begin{array}{c||c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\mathcal{L}_4\\
\hline\hline
\mathcal{R}_1&1,12321&12,1232&123&1234\\
\hline
\mathcal{R}_2&21,2321&2,232&23&234\\
\hline
\mathcal{R}_3&321&32&3,323&34,3234\\
\hline
\mathcal{R}_4&4321&432&43,4323&4,43234\\
\end{array}$$

The exceptional Coxeter type $H_3$ is the following: $$\xymatrix{1\ar@{-}[r]^{5}&2\ar@{-}[r]&3}\qquad \qquad 
\begin{array}{c||c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3\\
\hline\hline
\mathcal{R}_1&1,121&12,1212&123,12123\\
\hline
\mathcal{R}_2&21,2121&2,212&23,2123\\
\hline
\mathcal{R}_3&321,32121&32,3212&3,32123
\end{array}$$

The exceptional Coxeter type $H_4$ has the Coxeter diagram $$\xymatrix{1\ar@{-}[r]^{5}&2\ar@{-}[r]&3\ar@{-}[r]&4}$$ and the corresponding table looks as follows: $$\begin{array}{c||c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\mathcal{L}_4\\
\hline\hline
\mathcal{R}_1&1,121&12,1212&123,12123&1234,121234\\
\hline
\mathcal{R}_2&21,2121&2,212&23,2123&234,21234\\
\hline
\mathcal{R}_3&321,32121&32,3212&3,32123&34,321234\\
\hline
\mathcal{R}_4&4321,432121&432,43212&43,432123&4,4321234
\end{array}$$

The general dihedral type with Coxeter diagram $$\xymatrix{1\ar@{-}[r]^{k}&2},$$ where $k\geq 3$, looks as follows: $$\begin{array}{c||c|c}
&\mathcal{L}_1&\mathcal{L}_2\\
\hline\hline
\mathcal{R}_1&1,121,\dots,12\cdots 21&12,1212,12\cdots 12\\
\hline
\mathcal{R}_2&21,2121,\dots, 21\cdots 21&2,212,\dots,21\cdots 12
\end{array}$$ Note that the length of all elements in the latter table 