  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.
