Say I take an element of $\mathfrak{so}(3)$ with coordinates $(x,y,z)$ where $z$ is the element of the lie algebra dual which is dual to rotations about the $z$ axis etc.
If I the pull back of this form to $T_g^*SO(3)$ by left actions of $SO(3)$ on itself, is there an expression for $L_{g^{-1}}^*(z)$ in the traditional coordinate system on $T^*SO(3)$? That is, I want an expression $L_{g^{-1}}^*(z)=f_1\partial_{q_1}+f_2\partial_{q_2}+f_3\partial_{q_3}$ for ($q_1,q_2,q_3$) (some) coordinate system on $SO(3)$. I tried to do it with Euler angles and failed, I was thinking perhaps considering $SO(3)$ as the unit quaternions might be better but I don't even have a good expression for the tangent space...
This is starting to drive me a little crazy, so if anyone even had a hint on the easiest way to approach this it would be wonderful.