I am trying to understand the classic paper by Atiyah, Hitchin, and Singer
https://www.jstor.org/stable/79638
and I'm getting stuck on part of the proof of proposition 3.1. The proposition is determining the conditions for involutivity of the vector bundle $V(\bar{D})$ associated with a differential operator $\bar{D}=\sigma\nabla$, spanned by one forms $\theta_i$ and $\sigma_i$. In particular, it says that $d \theta_i$ and $d \sigma_i^\nu$ are sections of $V_2$ only if the curvature of $\nabla$ and $D_1 \sigma_i^\nu$ are sections of $V(S_2)$. Here $V_2$ is the image of $V(\bar{D})$ under exterior multiplication, and $V(S_2)$ is the image of the kernel of $\sigma$ under exterior multiplication. I don't understand why it is possible to conclude that the curvature and $D_1 \sigma_i^\nu$ must be sections of $V(S_2)$ instead of the presumably larger space $V_2$ (spanned by $V(S_2)$ in addition to two forms of the form $\theta_i\wedge \alpha_i$ for any $\alpha_i$), it is not really explained and not obvious to me.