I see different signs in Cartan's structure equation. For example, in Schoen-Yau 1981 proof of Positive Mass Theorem, it's stated as
$dw_a=w_{ab}\wedge w_b$
$dw_{ab}-w_{ac}\wedge w_{cb}=-\frac{1}{2}R_{abcd}w_c\wedge w_d$
while it's all plus sign in sources like https://planetmath.org/cartanstructuralequations
But Schoen-Yau seems to have same convention as Exercise 9, Chapter 30 of Modern Geometry by Dubrovin, Fomenko and Novikov. But I got sign difference with the claimed answer. It would be great if anyone can point out where I messed up.
Copying from the exercise: take $\{X_i\}$ ON basis, $\{\omega_i\}$ the dual basis. Here $\nabla_{X_k}X_j = \Gamma^{i}_{jk}X_i$ and $\omega_{ij}=\Gamma^{i}_{jk}\omega_k$, also $\Omega_{ij}=\frac{1}{2}\langle R(X_k,X_l)X_j,X_i\rangle \omega_k\wedge \omega_l$. Deduce
$d\omega_i=-\omega_j\wedge \omega_{ij}$
$d\omega_{ij}=\omega_{il}\wedge\omega_{lj}-\Omega_{ij}$
For the first equation I got $d\omega_i(X_k,X_l)=X_k(\omega_i(X_l))-X_l(\omega_i(X_k))-\omega_i([X_k,X_l])=-\omega_i(X_m)(\Gamma^{m}_{lk}-\Gamma^{m}_{kl})=-\Gamma^{i}_{lk}+\Gamma^{i}_{kl}$
while $\omega_{j}\wedge \omega_{ij} (X_k,X_l)=\omega_j(X_k)\omega_{ij}(X_l)-\omega_j(X_l)\omega_{ij}(X_k)=\Gamma^{i}_{kl}-\Gamma^{i}_{lk}$
So I got $d\omega_i = \omega_j\wedge \omega_{lj}$ which is of opposite sign to what claims to be answer.
Similarly for the second equation, I got oppsite sign for the RHS. I took $R(X_k,X_l)X_j=\big(\nabla_{X_k}\nabla_{X_l}-\nabla_{X_l}\nabla_{X_k}-\nabla_{[X_k,X_l]}\big) X_j$ (not sure whether the exercise assumed opposite sign).
Anywhere I understood the sign wrong? Thanks in advance!