I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is,
$h^*\Omega = 0$, where $\Omega$ is the curvature 2-form of the connection and $h^*$ is the map defined as
$h^*\beta(X_1,\dots,X_k)=\beta(hX_1,\dots,hX_k)$ where $hX_i$ are the horizontal components of the vector fields $X_i\in TP$.
I do not understand a line in the working on page 14
$d\left(\frac{1}{2}[\omega,\omega]\right)=\frac{1}{2}[d\omega,\omega]-\frac{1}{2}[\omega,d\omega]$
I am struggling to undersand 2 things:
1) How does the exterior derivative $d$ explicitly act on the 2 form $[\omega,\omega]$,
2) How does the 3 form $[d\omega,\omega]$ act on a triple of vector fields $(X,Y,Z)$?
I cannot seem to equate the two questions above given the definitions I have found from various resources on the internet, say, here