2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174021This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the classical root systems, $F_4$ and $E_6$. Such sequences are associated with reduced decompositions (words) in affine and non-affine Weyl groups. The existence of the non-gatherable triples is a combinatorial obstacle for using the technique of intertwiners for an explicit description of the irreducible representations of the (double) affine Hecke algebras, complementary to their algebraic-geometric theory.This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/Quantum AlgebraCombinatoricsRepresentation Theorytext