  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 digram. In types ${\bf DE}$, 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 ${\bf A}$. As an application, we describe the minimal
non-abelian ideals of $\mathfrak b$.
