Let $W$ be a Coxeter group with associated graph $G$.

Define $$X(G) = \{w \in W : \text{any two simple reflections} \,S_{\alpha}\, \text{and}\, S_{\beta} \,\text{appearing in any of the reduced expression of $w$ commutes} \} $$.

I have the following questions :

1. Is this set $X(G)$ studied in the literature?

2. What is the combinatorial significance of this set?

Thanks for your time and have a good day.