Let \({{\mathfrak {g}}}\) be a simple Lie algebra with a Borel subalgebra \({{\mathfrak {b}}}\). Let \(\Delta ^+\) be the corresponding (po)set of positive roots and \(\theta \) the highest root. A pair \(\{\eta ,\eta '\}\subset \Delta ^+\) is said to be glorious, if \(\eta ,\eta '\) are incomparable and \(\eta +\eta '=\theta \). Using the theory of abelian ideals of \({{\mathfrak {b}}}\), we (1) establish a relationship of \(\eta ,\eta '\) to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin diagram. In types \({{\mathbf {\mathsf{{{DE}}}}}}_{}\), we prove that if \(\{\eta ,\eta '\}\) corresponds to the edge through the branching node of the Dynkin diagram, then the meet \(\eta \wedge \eta '\) is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type \({{\mathbf {\mathsf{{{A}}}}}}_{}\). As an application, we describe the minimal non-abelian ideals of \({{\mathfrak {b}}}\).

令\（{{\ mathfrak {g}}} \）是具有Borel子代数\（{{\ mathfrak {b}}} \）的简单李代数。令\（\ Delta ^ + \）是相应的正根（po）集，而\（\ theta \）是最高根。一对\（\ {\ ETA，\ ETA“\} \子集\德尔塔^ + \）被说成是光荣，如果\（\ ETA，\ ETA '\）所无法比拟的和\（\ ETA + \ ETA' = \ theta \）。使用\（{{\ mathfrak {b}}} \）的阿贝尔理想理论，我们（1）建立\（\ eta，\ eta'\）的关系对于某些与长简单根相关的阿贝尔理想，（2）在光荣对和相邻的长简单根对之间提供自然双射（即，Dynkin图的某些边），并且（3）指出一个简单的变换连接对应于Dynkin图中入射边缘的两个光荣对。在类型\（{{\ mathbf {\ mathsf {{{DE}}}}} _ {} \）中，我们证明如果\（\ {\ eta，\ eta'\} \）对应于Dynkin图的分支节点，则Meet \（\ eta \ wedge \ eta'\）是唯一的最大非交换根。除类型\（{{\ mathbf {\ mathsf {{{{A}}}}}} __ {} \）之外，所有其他类型都具有此属性的类似物。作为应用程序，我们描述\（{{\ mathfrak {b}}} \）的最小非阿贝尔理想。