### Abstract

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(F_{4}) 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 F_{4}, symbolically written [F_{4}, F_{4}] = F_{4} + e_{7}F_{4}, describe the octonionic root system of E_{7} where the root system of the exceptional Lie algebra E_{7} are represented by the imaginary octonions. The automorphism group of the octonionic root system of E_{7} preserving the octonion algebra is the Chevalley group G_{2}(2) where the maximal subgroups of G_{2}(2) are the automorphism groups of the octonionic root systems of the maximal Lie algebras E_{6}, SU(2) x SO(12) and SU(8) of E_{7}. Another pairing of the quaternionic roots of the form [F _{4}, F_{4}] = F_{4} + σF_{4}, constitutes the root system of E_{8} in terms of icosians where half of the roots describe the roots of the noncrystallographic Coxeter group H_{4}. The relevance of H_{4} to E_{8} and the maximal subgroups of H _{4} have been discussed. Polyhedral structures obtained from the quaternionic root systems of H_{4} are described as the orbits of H _{3}.

Original language | English |
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Title of host publication | Mathematical Physics - Proceedings of the 12th Regional Conference |

Pages | 25-30 |

Number of pages | 6 |

Publication status | Published - 2007 |

Event | 12th Regional Conference on Mathematical Physics - Islamabad, Pakistan Duration: Mar 27 2006 → Apr 1 2006 |

### Other

Other | 12th Regional Conference on Mathematical Physics |
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Country | Pakistan |

City | Islamabad |

Period | 3/27/06 → 4/1/06 |

### Keywords

- Polyhedra
- Quaternions and octonions

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

*Mathematical Physics - Proceedings of the 12th Regional Conference*(pp. 25-30)