mathematical structure

Connection

$D = d + \mathbf{A} \wedge \,$ ,

where $d\,$ is the exterior derivative and $\wedge\,$ is the wedge product. Useful for later is the notation

$D = dx^\mu \partial_\mu \wedge + \mathbf{A}_\mu dx^\mu \wedge = dx^\mu (\partial_\mu + \mathbf{A}_\mu) \wedge = dx^\mu D_\mu \wedge\,$ .

$\mathbf{F} = D \mathbf{A} = d \mathbf{A} + \mathbf{A} \wedge \mathbf{A}\,$ ,

then

$\mathbf{F}\,$	$= dx^\mu D_\mu \wedge ( \mathbf{A}_{\nu} dx^\nu )\,$ ,
	$= ( D_\mu \mathbf{A}_{\nu} - D_\nu \mathbf{A}_{\mu} ) dx^\mu \otimes dx^\nu\,$ ,
	$= [ D_\mu , D_\nu ] dx^\mu \otimes dx^\nu\,$ .
	$= \frac{1}{2} [ D_\mu , D_\nu ] dx^\mu \wedge dx^\nu\,$ .

$D\mathbf{F}\,$	$= dx^\mu D_\mu \wedge ( \mathbf{F}_{\rho\sigma} dx^\rho \wedge dx^\sigma )\,$ ,
	$= (D_\mu \mathbf{F}_{\rho\sigma}) dx^\mu \wedge dx^\rho \wedge dx^\sigma\,$ ,
	$= [D_\mu , \mathbf{F}_{\rho\sigma}] dx^\mu \wedge dx^\rho \wedge dx^\sigma\,$ ,
	$= \frac{1}{2} [D_\mu , [ D_\rho, D_\sigma ] ] dx^\mu \wedge dx^\rho \wedge dx^\sigma\,$ ,
	$= 0\,$ , by virtue of the Jacobi identity $[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0\,$ .