A matrix representation of the automorphism group of pure integral octonions constituting the root system of E7 is constructed. It is shown that it is a finite subgroup of the exceptional group of G2 of order 12096, called the adjoint Chevalley group G2(2). Its four maximal subgroups of orders 432, 192, 192′ and 336 preserve, respectively, the octonionic root systems of E6, SO(12), SU(2)3SO(8) and SU(8). It is also shown explicitly that the full automorphism group of the pure octonions ±et (i=1,⋯, 7) constituting the roots of SU(2)7 is a group of order 1344. Possible implications in physics are discussed.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics