# Lie group and the frame bundle

$T(\mathbb{O}M)=H(\mathbb{O}M)\oplus V(\mathbb{O}M)$.
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 $f \in C^2(\mathbb{O}M)$.