Representation of a gauge field via intrinsic "BRST" operator

We show that there exists a representation of a matrix valued gauge field via intrinsic “BRST” operator assigned to matrix valued generators of a gauge algebra. In this way, we reproduce the standard formulation of the ordinary Yang - Mills theory. In the case of a generating quasigroup/groupoid, we give a natural counterpart to the Yang - Mills action. The latter counterpart does also apply as to the most general case of an involution for matrix-valued gauge generators.