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This question is a follow-up of the previous question 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 - an article which was also discussed on this site - 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 - modulo the fact that maybe not every element is a parabolic quasi-Coxeter element, cf. loc. cit., Condition 1.1 - 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.

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    $\begingroup$ Probably you want to forbid the identity element from being considered indecomposable. $\endgroup$ Jan 21 '20 at 18:16
  • $\begingroup$ The identity has order one, so, for the least common multiple, it does not matter. However, one is not a degree, as you said. $\endgroup$
    – user66288
    Jan 22 '20 at 3:40
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Concerning the question:

"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?"

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 $<7$.)

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