Quaternionic roots of  SO(8), SO(9), F4 and the related Weyl groups

Abstract:
The root systems of SO(8), F4 and the Coxeter group H4 can be respectively described by the discrete quaternions forming the binary tetrahedral group, binary octahedral group and binary icosahedral group. The relevance of the quaternionic representation of the binary icoshedral group to H4 has been extensively discussed in the literature. In this work we point out that there exist a natural  description of the root systems of SO(8), SO(9) and F4 and their Weyl groups by discrete quaternions. The triality of SO(8) manifests itself as  permutations of  three quaternionic imaginary units e1, e2 and e3. It has been shown that the relevant automorphism groups of the associated root systems are the finite subgroups of O(4) generated by the left-right actions on the root systems. The conjugacy classes of the Weyl groups follows from the conjugacy classes of the relevant quaternion groups. The relations between the Dynkin indices , standard orthogonal vector and the quaternionic weights have been obtained.