I have been struggling with relating two different forms of Bianchi's 2nd Identity: $d\Omega = \Omega\wedge\omega-\omega\wedge\Omega$ and
$\mathfrak{S}\left\{(\nabla_{Z}R)(X,Y,Z)+R(T(X,Y),Z)W)\right\}=0$
here $R$ and $T$ represent curvature and torsion tensor of a connection $\nabla$
$\Omega$ and $\omega$ represent curvature and connection form of $\nabla$
How do we relate the exterior derivative of curvature form to the covariant derivative of curvature and torsion tensors?
I am able to prove them independently but I am not able to prove the equivalence.
I am pretty new to differential geometry and I apologize if this question is very easy.
Here are some of my thoughts:
I started with curvature form. Suppose $X_1,X_2,X_3,...X_n$ represent the moving frames and $\theta^1, \theta^2,...,\theta_n$ represent the dual co frame. In the index notation the first form of bianchi's 2nd identity can be given as
$d\Omega^{i}_{j} = \Omega^i_k\wedge\omega^k_j-\omega^i_k\wedge\Omega^k_j$
(using einstien's summation convention)
Relation between cuvature tensor and curvature form can be given as
$R(X_k,X_l)X_j = \Omega^i_j(X_k,X_l)X_i$
in the index notation curvature tensor can be given as
$R(X_k,X_l)X_j = R^i_{jkl}X_i$
this gives us
$R^i_{jkl}X_i = \Omega^i_j(X_k,X_l)X_i$ or
$\Omega^i_{j} = \frac{1}{2}\sum_\limits{k,l}R^i_{jkl}\theta^k\wedge\theta^l$ this implies
$d\Omega^{i}_{j} =\frac{1}{2}\sum_\limits{m,k,l} \frac{\partial{R^i_{j,k,l}}}{\partial{X_m}}\theta^m\wedge\theta^k\wedge\theta^l$
the covariant derivative of R in index notation can be given as:
$\nabla R = \sum_\limits{h,j,k,l,i}R^i_{jkl;h}\theta^h\oplus\theta^j\oplus\theta^k\oplus\theta^l\oplus X_i$
where $R^i_{jkl;h}$ can be given as
$R^i_{jkl;h} = \frac{\partial{R^i_{j,k,l}}}{\partial{X_h}} + \sum\limits_{v=1}^n R^v_{j,k,l}\Gamma_{hv}^{i}-\sum_\limits{v=1}^{n}(R^i_{vkl}\Gamma^{v}_{hj}+R^i_{kvl}\Gamma^{v}_{hk}+R^i_{jkv}\Gamma^{v}_{hl})$
where $\Gamma$ can be given as
$\nabla_{X_i}{X_j} = \Gamma_{ij}^K X_k$
Also connection 1-form $\omega^i_{j}$ can be given as:
$\omega^i_j = \Gamma^i_{kj}\theta^k$
on one hand I have tensor product and on the other hand I have wedge product. At this point I am stuck and wonder if there is an alternate approach which doesn't involve coordinate representations. I am also not sure about my use of $X_i$'s as coordinate vector fields.