AUTOMORPHISM-GROUPS OF PURE INTEGRAL OCTONIONS

Matrix representation of the automorphism group of pure integral octon ions constituting the root system of E7 is constructed. It is shown th at it is a finite subgroup of the exceptional group of G2 of order 120 96, called the adjoint Chevalley group G2(2). Its four maximal subgrou ps of orders 432, 192, 192' and 336 preserve, respectively, the octoni onic root systems of E6, SO(12), SU(2)3 x SO(8) and SU(8). It is also shown explicitly that the full automorphism group of the pure octonion s +/-e(i) (i = 1, ..., 7) constituting the roots of SU(2)7 is a group of order 1344. Possible implications in physics are discussed.