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A principal connection is flat when its curvature vanishes. The identity $\mathcal{F}(X,Y)=-\mathcal{A}([X,Y])$ shows that this is equivalent to integrability of its horizontal distribution.

A principal connection on $P$ is a $G$-equivariant smooth splittingwhere $V_p$ is tangent to the $G$-orbit. Equivalently, it is a $\mathfrak{g}$-valued one-form $\mathcal{A}$ satisfying $\mathcal{A}({\xi }_P)=\xi$ and $R_g^{\ast }\mathcal{A}={\mathrm{Ad}}_{g^{-1}}\mathcal{A}$. Its curvature of a principal connection is

For a principal connection with connection form $\mathcal{A}$, the horizontal distribution of a principal connection is $H=\ker \mathcal{A}$, the complement of the tangent spaces to the $G$-orbits. Its curvature of a principal connection isIf $X,Y$ are horizontal vector fields, then $\mathcal{A}(X)=\mathcal{A}(Y)=0$, and henceThe Frobenius theorem says that $H$ is integrable exactly when $[X,Y]$ is horizontal for all horizontal $X,Y$. The displayed identity makes this equivalent to the vanishing of the horizontal two-form $\mathcal{F}$, hence to $\mathcal{F}=0$. Thus the horizontal distribution is integrable exactly for a flat principal connection.