Let $(W,S)$ be an arbitrary Coxeter system. We consider the following **scenario**:

Let $\mathcal{O}$ be a conjugacy class of an element $w$ in $W$ which is finite and which generates the whole group $W$.

The only **example** of the above scenario I know arises in simply laced irreducible finite Coxeter groups (i.e. all Coxeter integers are $\leq 3$) and for elements $w=s\in S$.

My question is: Are there any other examples, especially examples which occur in non simply laced Coxeter groups (i.e. with a Coxeter integer $\geq 4$), or examples which occur if $W$ is infinite?

This question is related. I found it by searching online. I impose finiteness of $\mathcal{O}$ on purpose to treat the case where $W$ is infinite, and this question tells us that a finite $\mathcal{O}$ can exist in this case.

**EDIT.** As I noticed meanwhile, I missed one example of the above scenario: The dihedral group $\mathbb{D}_m$ with $m\geq 5$ odd ($m$ is the unique Coxeter integer) and for elements $w=s\in S$ as in the first example. It seems I still had a false intuition concerning "non simply laced". Even if this second example is "non simply laced" according to the terminology, unlike Weyl groups, it only has one orbit on the root system / reflections. Sorry for the confusion. Let us focus on infinite $W$ from now on. We ask the

**QUESTION.** Does the above scenario occur for infinite $W$?

I think it can't. Proof (attempts), comments, thoughts, references are all welcome!

**Let me summarize some inputs of Nadeau and Speyer (thanks to both) and let me add clarification.** It holds that: $W$ is generated by one conjugacy class if and only if all elements of $S$ lie in one conjugacy class if and only if the subgraph of the Coxeter graph with odd labels and same vertices $S$ is connected.

This completely solves the question for finite Coxeter groups in the same sense as my text above. In the finite case, there are the simply laced Weyl groups (type $\mathsf{ADE}$) and $\mathbb{D}_m$ with $m$ odd - and only those.

As being said before the input, my question evolves around the infinite case where the above condition is necessary but not sufficient as the conjugacy classes of elements in $S$ are infinite by a result of Speyer linked above.