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

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.

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

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 $$<7$$.)