An inversion triple of an element w of a simply laced Coxeter group W is a set \(\{ \alpha , \beta , \alpha + \beta \}\), where each element is a positive root sent negative by w. We say that an inversion triple of w is contractible if there is a root sequence for w in which the roots of the triple appear consecutively. Such triples arise in the study of the commutation classes of reduced expressions of elements of W, and have been used to define or characterize certain classes of elements of W, e.g., the fully commutative elements and the freely braided elements. Also, the study of inversion triples is connected with the representation theory of affine Hecke algebras and double affine Hecke algebras. In this paper, we describe the inversion triples that are contractible, and we give a new, simple characterization of the groups W with the property that all inversion triples are contractible. We also study the natural action of W on the set of all triples of (not necessarily positive) roots of the form \(\{ \alpha , \beta , \alpha + \beta \}\). This enables us to prove rather quickly that every triple of positive roots \(\{ \alpha , \beta , \alpha + \beta \}\) is contractible for some w in W and, moreover, when W is finite, w may be taken to be the longest element of W. At the end of the paper, we pose a problem concerning the aforementioned action.

简单捆绑的Coxeter组W的元素w的倒数三元组是集合\（\ {\ alpha，\ beta，\ alpha + \ beta \} \），其中每个元素都是w派负的正根。我们说的三重反转W¯¯的缩小，如果有一个根序列W¯¯其中三联根连续出现。这样的三元组是在研究W元素的简化表示形式的换向类中产生的，并已用于定义或表征W元素的某些类例如，完全可交换元素和自由编织元素。同样，反三元组的研究与仿射Hecke代数和双仿射Hecke代数的表示理论有关。在本文中，我们描述了可收缩的三元组，并且给出了具有所有三元组都可收缩的性质的W组的新的简单特征。我们还研究了W对形式为\（\ {\ alpha，\ beta，\ alpha + \ beta \} \）的所有三重根（不一定是正数）的自然作用。这使我们能够相当快地证明，每三个正根\（\ {\α，\公测，\阿尔法+ \测试\} \）的缩小对一些W¯¯在W，而且，当W为有限时，w可以视为W的最长元素。在本文的最后，我们提出了有关上述动作的问题。