Given a metric, one can extract the vielbein 1-forms, which must then satisfy the Maurer-Cartan equations, which in the case of a torsionless space, are of the form

$$de^{a} + {\omega^a}_{b}\wedge e^{b} = 0 $$

where all indices are flat and ${\omega_\mu}^{ab}$ is the spin connection (with $\mu$ being a curved index).

Defining a vielbein 1-form as an object with just flat indices in Cadabra (until the Vielbein function is re-introduced), I would like to solve for ${\omega^a}_{b}$, the spin connection 1-forms. As I understand, I need the following ingredients:

- The wedge product.
- Handling differential forms, specifically introducing an operator $d$ which satisfies $d^2 = 0$ and $d(A_p \wedge B_q) = dA_p \wedge B_q + (-1)^p A_p \wedge dB_q$.
- The ability to go back and forth between coordinate differentials in the "curved" space, and vielbein 1-forms.
- Actually solving the Maurer-Cartan equatons.

Of course, one might argue that since there's an explicit formula for the spin connection components, why not just use it? I feel like it might be more efficient to get the 1-forms in flat space, rather than the components themselves. Any suggestions?

Also, is there any faster way in Cadabra to get the spin connection 1-forms? In the practical application I'm interested in, the metric splits into 3 parts.