*This question is a follow-up of the [previous question](https://mathoverflow.net/questions/343118/the-maximal-order-of-an-element-in-a-coxeter-group) and especially the last comment therein.*


Let $(W,S)$ be a finite Coxeter system with reflections $T$. Let $\ell\_T$ be the reflection length. According to [1611.03442v1, p. 2](https://arxiv.org/abs/1611.03442v1) - an article which was also [discussed on this site](https://mathoverflow.net/questions/249264/generalization-of-cycle-decomposition-to-coxeter-groups) - an element $x\in W$ is called indecomposable if there exists no nontrivial decomposition $x=uv$ such that $uv=vu$ and $\ell\_T(x)=\ell\_T(u)+\ell\_T(v)$. 


Is it true that the set of orders of indecomposable elements in $W$ coincides (with or without repitition) with the set of all degrees of $W$, and that - as a consequence of this and the generalized cycle decomposition in [loc. cit., Theorem 1.3](https://arxiv.org/abs/1611.03442v1) - modulo the fact that maybe not every element is a parabolic quasi-Coxeter element, cf. [loc. cit., Condition 1.1](https://arxiv.org/abs/1611.03442v1) - that - the least common multiple of all degrees equals the smallest positive integer $N$ such that $g^N=1$ for all $g\in W$?


**Remark.** I checked this for the symmetric group and some other rank two cases.


**Remark.** The Coxeter element itself is indecomposable (as its associated parabolic subgroup is the whole of $W$), has reflection length $|S|$, and its order is the Coxeter number which is known to be a degree. Further, every simple reflection is indecomposable of relfection length one, and its order is two, which is a degree as well. These observations are in favor of the question.



The least common multiple of all degrees of a finite Coxeter group and indecomposable elements in the generalized cycle decomposition
Concerning the question:


"Is it true that the set of orders of indecomposable elements in coincides (with or without repitition) with the set of all degrees of ?"


Thomas Gobet showed that in a finite irreducible Coxeter system each quasi-Coxeter element is indecomposable (see Proposition 3.5 in <https://arxiv.org/pdf/1611.03442.pdf>).


Carter classified the conjugacy classes in finite Weyl groups (see <http://www.numdam.org/article/CM_1972__25_1_1_0.pdf>). For example, the conjugacy class $E\_7(a\_3)$ in Carter's notation is a conjugacy class of a quasi-Coxeter element in a Coxeter system of type $E\_7$. The order of a corresponding quasi-Coxeter element is $30$. But $30$ is not a degree in type $E\_7$.
(I expect there to be a counterexample in rank $<7$.)


