Cartan first structure equation with input-first indices ($d{\varphi }_j={\sum }_i{\varphi }_i\wedge {\omega }_{ij}+{\tau }_j$)

If ${\nabla }_X{e}_i={\sum }_j{\omega }_{ij}(X)e_j$ and ${\varphi }_i$ is the dual coframe, the first index of $\omega$ is the input frame vector. Expand the connection derivative of $Y={\sum }_i{\varphi }_i(Y)e_i$ and antisymmetrize in $X,Y$. The formula for the exterior derivative of a one-form evaluated on vector fields gives ${\tau }_j=d{\varphi }_j-{\sum }_i{\varphi }_i\wedge {\omega }_{ij}$. This is the first structure equation with the indicated index convention. Reversing the wedge order changes the sign.