In a recent article, Cumplido and Paris studied the question of commensurability between Artin groups of spherical type. Their analysis left six cases undecided, for the following pairs of Artin groups: $(F_4,D_4)$, $(H_4,D_4)$, $(F_4,H_4)$, $(E_6,D_6)$, $(E_7,D_7)$, and $(E_8,D_8)$. In this note we resolve the first two of these cases, namely, we show that the Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$. As a key step, we realize the abstract commensurator of the Artin group of type $D_4$ as the extended mapping class group of the torus with three punctures. We also find the automorphism group of the Artin group of type $D_4$ and obtain a description of torsion elements, their orders and conjugacy classes in all irreducible Artin groups of spherical type modulo their centers.

在最近的一篇文章中，Cumplido 和 Paris 研究了球形阿廷群之间的可公度问题。他们的分析留下了六个未确定的案例，分别是以下几对 Artin 组：$(F_4,D_4)$, $(H_4,D_4)$, $(F_4,H_4)$, $({乙}_6,D_6)$, $({乙}_7,D_7)$， 和 $({乙}_8,D_8)$. 在本笔记中，我们解决了这些情况中的前两种情况，即，我们证明了 Artin 类型组$F_4$ 和 $H_4$ 与类型不相称 $D_4$. 作为关键的一步，我们实现了类型的Artin群的抽象公称$D_4$作为具有三个穿孔的环面的扩展映射类组。我们还找到了类型的 Artin 群的自同构群$D_4$ 并获得对所有不可约的球型 Artin 群中的扭转元素、它们的阶和共轭类的描述，以它们的中心为模。