I've been reading about quaternionic contact structures, but struggling to understand how they can be thought of as a G-structure (in this case $G=Sp(n)Sp(1)$).
For context, the definition I'm taking for a quaternionic contact structure is a Riemannian manifold $(M^{4n+3},g)$ equipped with a codimension-3 distribution $H\subset TM$, where $H$ is locally given as the kernel of three 1-forms $H= \ker\alpha_1\cap\ker\alpha_2\cap\ker\alpha_3$ such that $\frac{1}{2}d\alpha_1\rvert_H,\frac{1}{2}d\alpha_2\rvert_H,\frac{1}{2}d\alpha_3\rvert_H$ (viewed as skew endomorphisms using the metric) are a quaternionic structure on $H$.
According to (http://www.unm.edu/~vassilev/qc-form.pdf), one now defines 2-forms $\omega_1,\omega_2,\omega_3$ by extending the $\frac{1}{2}d\alpha_i\rvert_H$ by zero on $TM/H$, and it says that there is an $Sp(n)Sp(1)$ structure on the distribution $H$ defined by the 4-form $$ \Omega:=\omega_1\wedge\omega_1 + \omega_2\wedge\omega_2 +\omega_3\wedge \omega_3$$ (Is there an easy way to see that the stabilizer of $\Omega$ is $Sp(n)Sp(1)$?)
I'm confused why they say that the $Sp(n)Sp(1)$-structure is on the distribution $H$, rather than the whole tangent bundle. I'm also confused about the motivation behind defining $\Omega$--why should the frames which preserve it be "special"?
Any insight greatly appreciated!
