I am working in a (psuedo) Riemannian manifold with the fibres of orthonormal frame based on some Lie group. From the Levi-Civita connection, I can separate the tangent space of the frame bundle into horizontal and vertical bits, i.e.
T(\mathbb{O}M)=H(\mathbb{O}M)\oplus V(\mathbb{O}M).

Here is my problem, I have equations involving X_i f(\sigma) on the Lie group where the X_i s are a basis for the Lie algebra and f \in C^2(G). I want to be able to map these across to the frame bundle such that I get V_i f(s) where the V_i s are the canonical vertical vector fields and f \in C^2(\mathbb{O}M).

I have found that there exists a one-form between the tangent of the frames and the algebra which maps the horizontal bits to 0, but does this induce a map between the frame bundle and G? How?

Any suggestions would be welcome as my geometry knowledge is slim to none.