Non-associative Gauge Theory

We present a construction of gauge theory which its structure group is not a Lie group, but a Moufang loop which is essentially non-associative. As an example of non-associative algebra, we take octonions with norm one as a Moufang loop, with which we can produce an octonionic gauge theory. Our octonionic gauge theory is a natural generalization of Maxwell U(1)= S^1 gauge theory and Yang-Mills SU(2)= S^3 gauge theory. We also give the BPST like instanton solution of our octonionic gauge theory in 8 dimension.