Covariant wave equation for the Yang-Mills field strength ($D^{\rho }{D}_{\rho }{F}_{\mu \nu }=2[F_{\mu }{}^{\rho },F_{\nu \rho }]$)

Apply $D^{\rho }$ to the gauge-theory Bianchi identity and commute it through the other derivatives using $[D_{\mu },D_{\nu }]X=[F_{\mu \nu },X]$. The Yang-Mills equations remove the differentiated divergences. The two remaining commutator terms both equal $[F_{\mu }{}^{\rho },F_{\nu \rho }]$, proving the displayed nonlinear covariant wave equation for the Yang-Mills field strength on flat spacetime.

// Target: special-relativity.bigb