In Maxwell (electromagnetism) equation, U(1) gauge group is used, and U(1) is abelian, so it doesn’t require Yang-Mills theory.

However, What if the internal symmetry is non-Abelian (like $SU(2)$SU(2))? The gauge field has multiple components: The curvature picks up A∧A; Gauge bosons interact with each other. This is Yang–Mills theory.

Here is the Yang-Mills Equation