use the following general approach:

-   check that all elements given in the example have a unique reduced expression;

-   check that, making any step outside the list in the example, gives an element with more than one reduced expression;

-   note that all elements in a left cell start with the same simple reflection on the right, and all elements in a right cell start with the same simple reflection on the left;

-   use Proposition \[propJKL\].

We start with types $A_3$ and $D_4$. $$\xymatrix{1\ar@{-}[r]&2\ar@{-}[r]&3}\qquad \qquad 
\begin{array}{c||c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3\\
\hline\hline
\mathcal{R}_1&1&12&123\\
\hline
\mathcal{R}_2&21&2&23\\
\hline
\mathcal{R}_3&321&32&3
\end{array}$$ $$\xymatrix{&4&\\1\ar@{-}[r]&2\ar@{-}[r]\ar@{-}[u]&3}\qquad \qquad 
\begin{array}{c||c|c|c|c}
&\mathcal{L}_1&\mathcal{L}_2&\mathcal{L}_3&\mathcal{L}_4\\
\hline\hline
\mathcal{R}_1&1&12&123&124\\
\hline
\mathcal{R}_2&21&2&23&24\\
\hline
\mathcal{R}_3&321&32&3&324\\
\hline
\mathcal{R}_4&421&42&423&4
\end{array}$$

Similarly to the above, we have the following observation which follows directly from Proposition \[propJKL\].

\[corsimplylaced\] If the Coxeter diagram of $W$ is simply laced, then the cell $\mathcal{J}$ is strongly regular. Namely, the intersection $\mathcal{L}_i\cap \mathcal{R}_j$ consists of $s_js_{i_k}\dots s_{i_2}s_{i_1}s_i$, where $j-i_k-\dots- i_1-i$ is the unique path in the diagram connecting $i$ and $j$.

Because of Corollary \[corsimplylaced\], in the simply laced case, any simple transitive $2$-representation of $\underline{{\sc\mbox{S}\hspace{1.0pt}}}$ is equivalent to a cell $2$-representation by [@MM5 Theorem 18]. We note that the original formulation of [@MM5 Theorem 18] has 