In the note 1 of the Wikipedia Exterior covariant derivative article, it mentions some assertion that is crucial to the proof, but doesn't give a proof of the assertion itself. The assertion is something equivalent to the following.
Assertion: $(P,π,M)$ is a principal $G$-bundle with a given principal connection. $X$ is a fundamental vector field on $P$, $Y$ is a vector field on $M$, $Z$ is the horizontal lift of $Y$ by the principal connection. Then there is $[X,Z]=0$.
Can this be proved? I'm not sure whether $Z$ has to be a horizontal lift instead of just any horizontal vector field on $P$ for the above to be true.