Is it possible to take the Lie derivative with respect to a dual vector (1-form) field?
Suppose I have a vector field $X$ and two dual vector fields $\omega$ and $\sigma$. Then the Lie derivative with respect to a vector field, such as $\mathcal L_X\omega$, is standard. But what about $\mathcal L_\omega X$ or $\mathcal L_\omega\sigma$? I would prefer a coordinate expression.
(My motivation is a condition for a dual frame/tetrad/basis to be a coordinate/holonomic basis, analogous to the condition $\mathcal L_{\mathbf e_\alpha}\mathbf e_\beta=0$ for a vector basis $\{\mathbf e_\alpha\}$.)
There is a similar question here.