  Contact structures, as well as their holomorphic and quaternionic
counterparts are the primary examples of strongly bracket generating (or fat)
distributions. In this article we associate a numerical invariant to corank $2$
fat distribution on manifolds, referred to as \emph{degree} of the
distribution. The real distribution underlying a holomorphic contact structure
is of degree $2$. Using Gromov's sheaf theoretic and analytic techniques of
$h$-principle, we prove the existence of horizontal immersions of an arbitrary
manifold into degree $2$ fat distributions and the quaternionic contact
structures. We also study immersions of a contact manifold inducing the given
contact structure.
