Sultan Qaboos University,

College of Science, Department of Physics,

P.O.Box 36, 127 Al-Khoud, Muscat, OMAN

E-mail: kocam@squ.edu.om

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

**Abstract:**

The root systems of *SO*(8), *F*4 and the Coxeter group *H*4
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 *H*4 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 *F*4 and their
Weyl groups by discrete quaternions. The triality of SO(8) manifests itself
as permutations of three quaternionic imaginary units
*e*1,
*e*2 and *e*3. 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.