** Authors:**
Mehmet KOCA, Nazife Özdeş KOCA, Ramazan KOÇ

** Abstract: **
The lattice matching of two sets of quaternionic roots of $F_4$ leads to quaternionic roots of $E_8$ which has a decomposition $H_4 + \sigma H_4$ where the Coxeter graph $H_4$ is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of $H_3$ which has a natural extension to $H_3 + \sigma H_3$ describing the root system of the Lie algebra $D_6$. It is noted that there exist three lattices in 6-dimensions whose point group $W(D_6)$ admits the icosahedral symmetry $H_3$ as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group $H_2$ of order 10 is the Weyl group $W(A_4)$ where $H_2 + \sigma H_2$ constitute the root system of the Lie algebra $A_4$. The relevance of these systems to quasicrystals are discussed.

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