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


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


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.



Elements of Coxeter group whose simple reflections pairwise commute
If I see it correctly, there is not much going on in the set $X(G)$ from the viewpoint of Coxeter systems: Let $S$ be the simple system and let $G$ be the Coxeter graph of $(W,S)$ with vertex set $S$. Then
$$X(G) = \big\{ s\_1 \cdots s\_k \in W \mid \{s\_1,\ldots,s\_k\} \subseteq S\text{ with } s\_is\_j = s\_js\_i \text{ for all } 1 \leq i<j \leq k\big\}.$$
Since different choices of $s\_1,\ldots,s\_k$ yield different elements, $X(G)$ is in bijection with totally disconnected subsets of $G$.
This is, $\mathcal{D}(G) = \{ A \subseteq G \mid A \text{ finite and totally disconnected}\}$ and $\mathcal{D}(G) \rightarrow X(G)$ given by $A = \{ s\_1, \ldots s\_k\} \subseteq G$ of size $k$ is sent to $s\_1\cdots s\_k \in X(G) \subseteq W$ is a bijection.
In particular, $X(G)$ is finite for finite $S$ even if $W$ is infinite.


Observe that I used that any two reduced words have the same simple generators involved. Thus, your desired property means that your element lives in a standard parabolic subgroup of type $\mathbb{A}\_1^k$ for some $k$. This is exactly the describtion I gave.


