Cartan's second structural equation

See @lee2006riemannian exercise 7-2 or @Morita theorem 5.21.
Consider a Riemannian manifold $(M, g)$ . The Levi-Civita connection is described, in a local frame ${e_{1}, \dots, e_{n}}$ , by the connection forms $Θ$ . This connection is a connection on a fiber bundle , so it has a curvature which, expressed in the local frame, turns out to be

Ω := d Θ - Θ \land Θ

i.e., the curvature 2-forms.
On the other hand, the metric induces the Riemann curvature tensor $R$ .

Theorem (Cartan's second structural equation). It is satisfied that

R (X, Y) (e_{i}) = Ω_{i}^{j} (X, Y) e_{j} .

Written in components we have

R_{i a b}^{j} = Ω_{i}^{j} (e_{a}, e_{b}) .

$◼$
Proof:
According to the definition of the Riemann curvature tensor

R (X, Y) (e_{i}) = \nabla_{X} \nabla_{Y} e_{i} - \nabla_{Y} \nabla_{X} e_{i} - \nabla_{[X, Y]} e_{i} .

and denoting by $Θ$ the connection 1-forms, we have

R (X, Y) (e_{i}) = \nabla_{X} [Θ_{i}^{j} (Y) e_{j}] - \nabla_{Y} [Θ_{i}^{j} (X) e_{j}] - Θ_{i}^{j} ([X, Y]) e_{j} =

= X (Θ_{i}^{j} (Y)) e_{j} + Θ_{i}^{j} (Y) \nabla_{X} e_{j} - Y (Θ_{i}^{j} (X)) e_{j} - Θ_{i}^{j} (X) \nabla_{Y} e_{j} - Θ_{i}^{j} ([X, Y]) e_{j} .

R (X, Y) (e_{i}) = d Θ_{i}^{j} (X, Y) e_{j} + Θ_{i}^{j} (Y) \nabla_{X} e_{j} - Θ_{i}^{j} (X) \nabla_{Y} e_{j}

And then

R (X, Y) (e_{i}) = d Θ_{i}^{j} (X, Y) e_{j} + Θ_{i}^{j} (Y) Θ_{j}^{k} (X) e_{k} - Θ_{i}^{j} (X) Θ_{j}^{k} (Y) e_{k} =

and renaming indices

R (X, Y) (e_{i}) = d Θ_{i}^{j} (X, Y) e_{j} - Θ_{i}^{k} \land Θ_{k}^{j} (X, Y) e_{j} =

= (d Θ_{i}^{j} (X, Y) + Θ_{k}^{j} \land Θ_{i}^{k} (X, Y)) e_{j}

And by definition of curvature 2-forms

R (X, Y) (e_{i}) = Ω_{i}^{j} (X, Y) e_{j} .

Remarks

In a flat space $d Θ = - Θ \land Θ$ .
Related to the Maurer-Cartan equation, see Generalization of the flatness of R3.