Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. Let $Δ^+$ be the corresponding (po)set of positive roots and $θ$ the highest root. A pair $\{η,η'\}\subset Δ^+$ is said to be glorious, if $η,η'$ are incomparable and $η+η'=θ$. Using the theory of abelian ideals of $\mathfrak b$, we (1) establish a relationship of $η,η'$ 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 digram. In types ${\bf DE}$, we prove that if $\{η,η'\}$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $η\wedgeη'$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type ${\bf A}$. As an application, we describe the minimal non-abelian ideals of $\mathfrak b$.