I'm reading the book instantons and 4-manifolds written by Uhlenbeck and Freedman. I meet a problem relatived to homogeneous space but I know nothing about it.
It's quite a specific problem. The book says that since $Sp(2)$ acts by isometrics on $S^7$, the vertical tangent space $VTS^7_{(q^1,q^2)}={(pq^1,pq^2)}: p\in Im \mathbb{H}$ defines a homogeneous horizontal distribution on $S^7$. And the correspondent connection is $Im(q^1d\bar{q}^1+q^2d\bar{q}^2)$.
My question is, how to derive the connection form from the argument above? Thanks in advance！