Let $A$ be a *flat* connection on a principal $G$-bundle $G\hookrightarrow P\to M$.
Consider an *homotopically trivial* loop $\gamma \subset M$. For simplicity, suppose $\gamma = \partial D$ is the boundary of an smoothly embedded disk.
I was trying to prove that the holonomy of $A$ along $\gamma$ is trivial (must be by flatness).
The holonomy of $A$ along $\gamma$ is given by
$$\exp^{-\int_\gamma A} \quad ( \ = 1 \text{ by flatness}) $$
Therefore applying Stokes, we get ($F_A=0$)
$${\int_\gamma A}=\int_D dA = \int_D (F_A - \frac 1 2 [A\wedge A])= \int_D - \frac 1 2 [A\wedge A]$$

Supposing that $G$ is not abelian, then I would like to understand why $\int_D [A\wedge A]$ lies in the kernel of $\exp$.

I know other proofs of this fact, e.g flatness implies the horizontal distribution is trivial hence we can use charts where $A$ is identically zero. But I hope there is a simpler explanation for this, i.e. we can prove that trivial loops have trivial holonomy just using that $F_A=0$ instead of the deeper/equivalent integrability of the horizontal distribution.