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JOURNAL OF MATHEMATICAL PHYSICS, vol.33, no.2, pp.497-510, 1992 (SCI-Expanded)
Octonionic root system of E8 is decomposed as the 9 sets of Hurwitz integers, each set satisfying the binary tetrahedral group structure, and the 12 sets of quaternionic units, each set obeying the dicylic group structure of order 12. This fact is used to prove that the automorphism group of the octonionic root system of E7 is the finite subgroup of G2, of order 12 096, an explicit 7 X 7 matrix realization of which is constructed. Possible use of the octonionic representation of the E6 root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested.