involutions in their cells follows directly from the definitions since the value of $\mathbf{a}$ on $\mathcal{J}$ is $1$, implying , . This completes the proof.

An important consequence of Proposition \[propJKL\] is the following: if $s$ and $t$ are different elements in $S$, then the longest element in the parabolic subgroup of $W$ generated by $s$ and $t$ has two different reduced expressions and thus does not belong to $\mathcal{J}$. We also refer the reader to [@De; @BW] for related results.

Examples and special cases {#s4.2}
--------------------------

Using Proposition \[propJKL\], one can describe elements in $\mathcal{J}$ explicitly as products of simple reflections. Here we list several examples and special cases under the following conventions:

-   we provide a picture of the Coxeter diagram followed by the list of elements in $\mathcal{J}$ organized in a square table, each element is given by its unique reduced expression;

-   vertices of the Coxeter diagram are numbered by positive integers and we write $i$ for the corresponding simple reflection $s_i$;

-   the left cell $\mathcal{L}_i$ containing $i$ is the $i$-th column of the diagram;

-   the right cell $\mathcal{R}_i$ containing $i$ is the $i$-th row of the diagram;

To verify these examples one could 