I've been reading Nakahara's "Geometry, Topology and Physics" and found something quite strange on the section 10.3.3 which discusses the geometrical meaning of the curvature of a connection. It is possible to find there the following text:
We first show that $\Omega(X,Y)$ yields the vertical component of the Lie bracket $[X,Y]$ of horizontal vectors $X,Y\in H_u P$. It follows from $\omega(X)=\omega(Y)=0$ that
$$d_P\omega(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y])=-\omega([X,Y]).$$
My problem here is the following. As I've previously studied in other books, the Lie bracket can only be computed for vector fields. That is, given a smooth manifold $M$, a point $a\in M$ and two vectors $v,w\in T_a M$ it is totally meaningless to talk about $[v,w]$. In that case, $[\cdot,\cdot]$ is only meaningful for vector fields.
On the text, Nakahara is talking about picking two vectors $X,Y\in H_u P$ the horizontal subspace of the tangent space $T_u P$ at $u\in P$ and then he is talking about the Lie bracket $[X,Y]$.
And this is not the only place where he does that. Some paragraphs later we can see the same thing being done again, so it is certainly not a typo.
Is that really a mistake in the book? Or is there something I'm missing? Perhaps there is some natural extension of the vectors to vector fields when dealing with connections on principal bundles and I'm not aware of that.
Am I missing something or the book has a mistake in it?
You are right, but, the paragraph you quoted shows that the vertical component of $[\tilde{X}, \tilde{Y}]$, for any horizontal "extensions" $\tilde{X}$, $\tilde{Y}$ of $X$, $Y$ respectively is the same, being equal to $-F(X,Y)$, which only depends on the values $X$ and $Y$ at the point considered (and thus independent of the choices of horizontal extensions $\tilde{X}$, $\tilde{Y}$).