Let $e_1,...,e_n$ be an oriented, orthonormal frame as before, $\theta^1,...,\theta^n$ the coframe and $\omega_1^1,....,\omega_n^n$ the connection forms. We have $d\theta^i=\sum_{k=1}^n\theta^k\omega_k^i$ for $i=1,...,n$. On one hand $d\theta^i(e_j,e_k)=e_j(\theta^ie_k)-e_k(\theta^ie_j)-\theta^i([e_j,e_k])=-\theta^i([e_j,e_k])$, on the other hand $(\sum_{l=1}^n\theta^l\wedge\omega_l^i)(e_j,e_k)=\sum_{l=1}^n(\delta_{lj}\omega_l^i(e_k)-\delta_{lk}\omega_l^i(e_j))=\omega_j^i(e_k)-\omega_k^i(e_j)$. Since the frame is orthonormal, $\theta^i=\langle e_i,-\rangle$, so we obtain $\langle e_i,[e_j,e…