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

The root systems of SO(8), SO(9) and F₄ are constructed by quaternions. Triality manifests itself as permutations of pure quaternion units e₁, e₂ and e₃. It is shown that the automorphism groups of the associated root systems are the finite subgroups of O(4) generated by left-right actions of unit quaternions on the root systems. The relevant finite groups of quaternions, the binary tetrahedral and binary octahedral groups, play essential roles in the construction of the Weyl groups and their conjugacy classes. The relations between the Dynkin indices, standard orthogonal vector and the quaternionic weights are obtained.