### Abstract

The octonionic root system of the exceptional Lie algebra E_{8} has been constructed from the quaternionic roots of F_{4} using the Cayley-Dickson doubling procedure where the roots of E_{7} correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E_{7} is the adjoint Chevalley group G_{2}(2) of order 12096. One of the four maximal subgroups of G_{2}(2) of order 192 preserves the quaternion subalgebra of the E_{7} root system. The other three maximal subgroups of orders 432; 192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E_{6} × U(1), SU(2) × SO(12) and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

Original language | English |
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Pages (from-to) | 808-823 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 422 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Apr 15 2007 |

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### Keywords

- Group structure
- M-Theory
- Quaternions
- Subgroup structure

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

_{2}(2) of order 12096 and the octonionic root system of E

_{7}

*Linear Algebra and Its Applications*,

*422*(2-3), 808-823. https://doi.org/10.1016/j.laa.2006.12.011