\begin{equation}

[D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi,

\end{equation}

where the scalar field is valued in the lie algebra of a Yang-Mills theory. Here,

\begin{equation}

D_{\mu}=\partial_{\mu} + [A_{\mu},\Phi],

\end{equation}

and

\begin{equation}

F_{\mu,\nu}=\partial_{\mu} A_{\nu} - \partial_{\nu}A_{\mu} + [A_{\mu}, A_{\nu}].

\end{equation}

When the scalar field is an n-tuple, so that the second term in the covariant derivate does not have the brackets, I get the correct result. But here, my calculations give

\begin{equation}

[D_{\mu},D_{\nu}]\Phi=\left(\partial_{\mu} A_{\nu}- \partial_{\nu}A_{\mu}\right)\Phi + \left[[A_{\mu}, \cdot ] , [A_{\nu}, \cdot]\right]\Phi

\end{equation}

The problem is that I don't see how the last term of this equation could become the remaining part of the Field Strength, and I don't see any other possibility! Expanding it, I find

\begin{equation}

\left[[A_{\mu}, \cdot ] , [A_{\nu}, \cdot]\right]\Phi = [A_{\mu}, [A_{\nu}, \Phi]] + [A_{\nu}, [\Phi, A_{\mu}]],

\end{equation}

which, by Jacobi's identity, can be written

\begin{equation}

\left[[A_{\mu}, \cdot ] , [A_{\nu}, \cdot]\right]\Phi =[[A_{\mu},A_{\nu}],\Phi],

\end{equation}

and that's the closest I get. What is wrong with this result? Is it somehow equivalent to the right one? If so, how?