Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.
I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence
$$ 0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow T M\rightarrow 0$$
where $\text{ad}(E_G)$ is the adjoint vector bundle for $E_G$ and $\text{At}(E_G)$ is the Atiyah bundle for $E_G$.
Reference given for this is Atiyah's paper. This paper is slightly difficult to read.
Can some one give an outline of this construction or point out some exposition where this is written in detail.