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I'm looking at control systems on SO(3) of the form $\dot{g} = gf(g) + gh(g)u$, where $gf(g)$ is the drift vector field and $gh(g)$ is the control vector field. I'm interested in expressing these fields in terms of some local coordinates, i.e.

$$gh(g) = \sum_{i}X^i\frac{\partial}{\partial x^i}$$

How would I go about computing the $X^i$s?

As a specific example, suppose $h = L_x$ (an element of the standard basis of $\mathfrak{so}(3)$), and we use geodesic polar coordinates with the exponential map as the chart function. What would the local coordinate representation of $gh$ look like?

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