Octonionic gauge theory from spontaneously broken SO(8)

An attempt is made to construct a gauge theory based on a bimodular representation of the octonion algebra, the non-associativity of which is manifested as a non-closure of the bimodule algebra. It is found that this fact leads to gauge-non-invariance of the theory. However, the bimodule algebra can be embedded in SO(8), the gauge theory of which can be broken down to give a massless SO(7) theory together with a massive octonionic gauge theory.