= Solution
Work over a coordinate neighborhood $U\subset B$ with coordinates $x^i$, and choose a basis $T_a$ of the <Lie algebra>, with $[T_a,T_b]=c^c{}_{ab}T_c$. Put $A=A_i^aT_a\,dx^i$. For each $T_a$ let $R_a$ be its <right-invariant vector field> on the fiber group. In matrix notation $R_a(\gamma)=T_a\gamma$, and
$$
[R_a,R_b]=-c^c{}_{ab}R_c.
$$
This minus sign is the <right-invariant vector fields realize the opposite Lie algebra> rule. It is essential here. Define the <coordinate horizontal lifts of a principal connection> by
$$
\boxed{H_i=\partial_i-A_i^aR_a.}
$$
Their projections are $\pi_*H_i=\partial_i$, so they are linearly independent and there are exactly $\dim B$ of them. Also $d\gamma(H_i)=-A_i\gamma$, whence
$$
\omega(H_i)=\gamma^{-1}A_i\gamma+\gamma^{-1}(-A_i\gamma)=0.
$$
They therefore span the <horizontal distribution of a principal connection> on this trivialization.
The coefficients $A_i^a$ depend only on the base variables, so taking <Lie brackets> gives
$$
[H_i,H_j]=-\left(\partial_iA_j^c-\partial_jA_i^c+c^c{}_{ab}A_i^aA_j^b\right)R_c=-F_{ij}^cR_c,
$$
where $F_{ij}=\partial_iA_j-\partial_jA_i+[A_i,A_j]$ and $F=\tfrac12F_{ij}\,dx^i\wedge dx^j=dA+A\wedge A$. Since the <right-invariant vector fields> form a basis on each fiber, \b[$\boxed{[H_i,H_j]=0\text{ for all }i,j\iff F=0}$]. This is the local coordinate version of vanishing <curvature of a principal connection> and, by the <Frobenius theorem>, integrability of the horizontal distribution.
The coordinate qualification is necessary: horizontal lifts of arbitrary noncommuting base fields need not commute even for a <flat principal connection>. Nor does flatness guarantee a globally defined coordinate frame or a global horizontal section; <holonomy of a connection> can obstruct the latter. If the requested frame were read globally, the trivial flat bundle $S^2\times G$ would already be a counterexample: restricting a global horizontal frame to the identity-fiber section would trivialize $TS^2$, contrary to the <Hairy ball theorem>. The construction on each coordinate trivialization supplies the intended result without that global claim.
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