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For a finite real reflection group, $W$, with Coxeter element $\gamma$ we give a uniform proof that the closed interval, $[I, \gamma]$ forms a lattice in the partial order on $W$ induced by reflection length. The proof involves the construction of a simplicial complex which can be embedded in the type W simplicial generalised associahedron.
Brady, T., & Watt, C. (2005). Lattices in finite real reflection groups. arXiv: Combinatorics. DOI: 10.21427/kzsd-1a33