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\begin{document}$${{\mathfrak {g}}}$$\end{document} be a simple Lie algebra with a Borel subalgebra b\documentclass[12pt]{minimal}
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\begin{document}$${{\mathfrak {b}}}$$\end{document}. Let Δ+\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^+$$\end{document} be the corresponding (po)set of positive roots and θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} the highest root. A pair {η,η′}⊂Δ+\documentclass[12pt]{minimal}
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\begin{document}$$\{\eta ,\eta '\}\subset \Delta ^+$$\end{document} is said to be glorious, if η,η′\documentclass[12pt]{minimal}
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\begin{document}$$\eta ,\eta '$$\end{document} are incomparable and η+η′=θ\documentclass[12pt]{minimal}
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\begin{document}$$\eta +\eta '=\theta $$\end{document}. Using the theory of abelian ideals of b\documentclass[12pt]{minimal}
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\begin{document}$${{\mathfrak {b}}}$$\end{document}, we (1) establish a relationship of η,η′\documentclass[12pt]{minimal}
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\begin{document}$$\eta ,\eta '$$\end{document} 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 DE\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbf {\mathsf{{{DE}}}}}}_{}$$\end{document}, we prove that if {η,η′}\documentclass[12pt]{minimal}
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\begin{document}$$\{\eta ,\eta '\}$$\end{document} corresponds to the edge through the branching node of the Dynkin diagram, then the meet η∧η′\documentclass[12pt]{minimal}
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\begin{document}$$\eta \wedge \eta '$$\end{document} is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type A\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbf {\mathsf{{{A}}}}}}_{}$$\end{document}. As an application, we describe the minimal non-abelian ideals of b\documentclass[12pt]{minimal}
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\begin{document}$${{\mathfrak {b}}}$$\end{document}.
