I saw in wikipedia (http://en.m.wikipedia.org/wiki/Curvature_form) the following identity for the curvature 2-form of a principal bundle $$2\Omega(X,Y)=h[X,Y]-[hX,hY]$$ where $X,Y\in T_uP$, $P$ being the principal bundle, and $hX$ is the horizontal component of $X$. The comment of the proof is also not very helpful:
"Proof: We can assume X, Y are horizontal (otherwise both side vanish). In that case, this is a consequence of the invariant formula for exterior derivative d and the fact ω(Z) is a unique Lie algebra element that generates the vector field Z."
1) Can someone help me by explaining me with more detail the proof of this fact or at least point me to some reference where I can find this identity?
2) I think this identity has something to do with the fact that the lie bracket of two horizontal vectors is a vertical vector (a fact I read in Nakahara's book "Geometry, topology and Physics", 2nd edition, page 388) but I'm also not able to prove this nor to follow Nakahara's proof. Can someone also help me with this?