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 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.