Let $P$ be a principal fiber bundle with structure group $G$ acting freely on the right. Let $T_uP = G_u + Q_u$ be a connection on $P$, where $u\in P$ and $G_u$ is the vertical space consisting of vectors tangent to the fibers.
Proposition 4.1 Let a Lie group $G$ act on $P$ on the right. The mapping $\sigma: \mathfrak{g}\mapsto \mathfrak{X}(P)$ which sends $A$ into $A^*$ is a Lie algebra homomorphism. If $G$ acts effectively on $M$, then $\sigma$ is an isomorphism of $\mathfrak{g}$ into $\mathfrak{X}(P)$.
The above proposition was taken from chapter 1 in Kobayashi's book Foundations of Differential Geometry.
Since every free action is effective (faithful), the map $\sigma$ is an isomorphism between the algebras $\mathfrak{X}(P)$ and $\mathfrak{g}$.
For $u\in P$ and $A\in\mathfrak{g}$, we have
$${A^*}_u := \left.\frac{d}{dt}\right\vert_{t=0} \left( u\cdot \exp(tA)\right).$$
So we conclude that $A^*$ is a vertical vector field, since it is tangent to the fiber through $u$.
Conclusion: by the above mentioned isomorphism, every vector field $X\in\mathfrak{X}(P)$ is a fundamental vector field associated to some $A\in\mathfrak{g}$, i.e, $X=A^*$. Therefore every vector field over $P$ is vertical.
Geometrically, this doesn't look right to me. I need help to find out where I did wrong.
Best Answer
The algebras $\mathfrak X(P)$ and $\mathfrak g$ cannot be isomorphic at all, since $\mathfrak g$ is finite dimensional whereas $\mathfrak X(P)$ is infinite dimensional.
The stated proposition in Kobayashi & Nomizu is wrong. Probably what the authors meant was that $\sigma$ is injective, eg. it is a Lie algebra isomorphism between $\mathfrak g$ and $\sigma(\mathfrak g)<\mathfrak X(P)$.
Instead what is an actual isomorphism is that for fixed $u\in P$, there is a map $\sigma_u:\mathfrak g\rightarrow G_u$, which is an isomorphism, though in this case it is just a linear isomorphism. It sends $A\in\mathfrak g$ into $$ A^\ast_u=\frac{d}{dt}u\cdot\exp(tA)|_{t=0}. $$
The vector field $(A^\ast)_u=A^\ast_u$ is everywhere vertical, so if we define the space $\mathfrak X^V(P)$ to be the space of all vertical vector fields on $P$, then the map $\sigma:\mathfrak g\rightarrow\mathfrak X^V(P)<\mathfrak X(P)$ is still an injective Lie algebra homomorphism (remember that since the vertical distribution is integrable, Lie brackets of vertical fields are themselves vertical), however even in this case, it fails to be an isomorphism, as if $X\in\mathfrak X^V(P)$ is a vertical vector field, then it is true that for any $u\in P$ there exists an $A_{X_u}\in\mathfrak g$ such that $X_u=\sigma_u(A_{X_u})$, however in general there is not one $A\in\mathfrak g$ such that $X_u=\sigma_u(A)$ for every $u\in P$.
If $X$ is special such that there does exist one $A\in\mathfrak g$ such that $X_u=\sigma_u(A)$ for every $u\in P$, then this is what Kobayashi and Nomizu refer to as a fundamental vector field.