Symmetries of the octonionic root system of E₈

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 G ₂, of order 12 096, an explicit 7×7 matrix realization of which is constructed. Possible use of the octonionic representation of the E ₆ root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested.