Octonions, G_2 Symmetry, Generalized Self-Duality and Supersymmetries in   Dimensions D \le 8
  We establish N=(1/8,1) supersymmetric Yang-Mills vector multiplet with generalized self-duality in Euclidian eight-dimensions with the original full SO(8) Lorentz covariance reduced to SO(7). The key ingredient is the usage of octonion structure constants made compatible with SO(7) covariance and chirality in 8D. By a simple dimensional reduction together with extra constraints, we derive N=1/8+7/8 supersymmetric self-dual vector multiplet in 7D with the full SO(7) Lorentz covariance reduced to G_2. We find that extra constraints needed on fields and supersymmetry parameter are not obtained from a simple dimensional reduction from 8D. We conjecture that other self-dual supersymmetric theories in lower dimensions D =6 and 4 with respective reduced global Lorentz covariances such as SU(3) \subset SO(6) and SU(2) \subset SO(4) can be obtained in a similar fashion. 
Hitoshi Nishino and Subhash Rajpoot