Let's consider the frame bundle $\text{Fr}(\mathbb{R}^n) = \mathbb{R}^n \times \text{GL}_n$ with bundle projection $\pi : \text{Fr}(\mathbb{R}^n) \to \mathbb{R}^n$ and tautological $1$-form $\theta \in \Omega^1(\text{Fr}(\mathbb{R}^n), \mathbb{R}^n)$ given in this trivialization by
$$\theta_{(x, g)}(A, B) = g^{-1} A$$
Consider, for a point $p \in \text{Fr}(\mathbb{R}^n)$, a horizontal subspace $H_p \subset T_p \text{Fr}(\mathbb{R}^n)$, that is, one for which $d_p \pi|_{H_p} : H_p \to T_{\pi(p)} \mathbb{R}^n$ is an isomorphism, such that $(d\theta)|_{H_p} = 0$ (that is, $d\theta$ is always zero when calculated at two vectors from $H_p$). I've been trying to show that in this case necessarily we'd have $H_p = \mathbb{R}^n \oplus 0$, but I can't seem to do it. (I expect this to be true by some general considerations but I might be mistaken). Can anyone see how to do it?
It's probably helpful to note that $\iota_{\widetilde{X}} d\theta = - X \cdot \theta$, where $\iota$ is contraction and $\widetilde{X}$ is the infinitesimal generator of some $X \in \text{gl}_n$ (this follows essentially from Cartan's magic formula).