Quaternionic and octonionic structures of the exceptional lie algebras

The Cayley-Dickson procedure has been used to construct the root systems of SO(8), SO(9) and the exceptional Lie algebra F4 in terms of quaternions. The Aut(F₄) has a simple presentation of the form (O,O) ⊕ (O,O)* where 0 represents the binary octahedral group. Two sets of quaternionic root system of F₄, symbolically written [F₄, F₄] = F₄ + e₇F₄, describe the octonionic root system of E₇ where the root system of the exceptional Lie algebra E₇ are represented by the imaginary octonions. The automorphism group of the octonionic root system of E₇ preserving the octonion algebra is the Chevalley group G₂(2) where the maximal subgroups of G₂(2) are the automorphism groups of the octonionic root systems of the maximal Lie algebras E₆, SU(2) x SO(12) and SU(8) of E₇. Another pairing of the quaternionic roots of the form [F ₄, F₄] = F₄ + σF₄, constitutes the root system of E₈ in terms of icosians where half of the roots describe the roots of the noncrystallographic Coxeter group H₄. The relevance of H₄ to E₈ and the maximal subgroups of H ₄ have been discussed. Polyhedral structures obtained from the quaternionic root systems of H₄ are described as the orbits of H ₃.