Slight confusion about projection operator in principal bundle Let $P(M,G)$ be a principal bundle with base space $M$ and structure group $G$. Let $\Gamma$ be a connection on $P$ so that the horizontal subspaces are denoted by $\Gamma_u$ for $u\in P$. Let $\pi:P\to M$ be the projection operator.
Let $A$ be a vertical vector field, i.e., tangent to the fibres, and $X$ be a horizontal vector field. Then we know that $[A,X]$ is also horizontal [1]. However, if we consider the pushforward induced by $\pi$, it would seem that $\pi [A,X]=[\pi A,\pi X]$. Since $\pi A=0$, we would have $[A,X]$ is vertical. What is wrong with the logic here?
[1] Kobayashi, lemma in Theorem 5.2 of Chapter II
EDIT. Just as a reminder for my own purpose. $[X,Y]_u$ depends on values on a neighborhood due to it definition $X_u(Yf)-Y_u(Xf)$, i.e., $Xf,Yf$ must be defined near the evaluation point $u$.
 A: You can only push-forward vector fields through a diffeomorphism. While you can push-forward a single tangent vector through an arbitrary smooth map as (that is just the differential), so to say taking the "pointwise push-forward" and in this way explain that a vector field is vertical if any of its values is vertical, you don't obtain a vector field. Simply let $\phi:M\to N$ be not surjective, then what should a push-forwarded vector field look like in a point which is not contained in the range of $\phi$. And you cannot solve this if you claim that $\phi$ is surjective (as e.g. $\pi$ is). If $\phi$ is not injective you run into other troubles, what should a push-forwarded vector field look like in a point which is hit by $\phi$ twice. Which possible value would you choose? And even claiming $\phi$ to be bijective does not resolve this, take $\phi:\mathbb{R}\to\mathbb{R},~x\mapsto x^{\frac{1}{3}}$. Can you see why you cannot push-forward a vector field through this map?
That is, your error lies in the usage of the formula
$$
\pi[A,X]=[\pi A,\pi X],
$$
the RHS makes no sense as the expression $[\pi A,\pi X]\big|_x$ does not only depend on $A\big|_x$ and $X\big|_x$ but also on $A$ and $X$ on an arbitrarily small neighborhood of $x$. So to apply this formula you need a diffeomorphism which $\pi$ surely is not (unless $G=\{e\}$, which I guess is not assumed).
