The Chevalley group G2(2) of order 12096 and the octonionic root system of E7

Abstract

The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quaternionic roots of F4 using the Cayley-Dickson doubling procedure where the roots of E7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E7 is the adjoint Chevalley group G2(2) of order 12096. One of the four maximal subgroups of G2(2) of order 192 preserves the quaternion subalgebra of the E7 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 E6 × 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. © 2006 Elsevier Inc. All rights reserved.