Turkish Journal of Physics
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.
KOCA, Mehmet; KOCA, Nazife Özdeş; and KOÇ, Ramazan (1998) "Quaternionic Roots of E 8 Related Coxeter Graphs and Quasicrystals," Turkish Journal of Physics: Vol. 22: No. 5, Article 8. Available at: https://journals.tubitak.gov.tr/physics/vol22/iss5/8