= Cartan first structure equation with input-first indices
{c}
{title2=$d\phi_j=\sum_i\phi_i\wedge\omega_{ij}+\tau_j$}
If $\nabla_Xe_i=\sum_j\omega_{ij}(X)e_j$ and $\phi_i$ is the dual <coframe>, the first index of $\omega$ is the input frame vector. Expand the connection derivative of $Y=\sum_i\phi_i(Y)e_i$ and antisymmetrize in $X,Y$. The formula for the <exterior derivative of a one-form evaluated on vector fields> gives $\tau_j=d\phi_j-\sum_i\phi_i\wedge\omega_{ij}$. This is the first structure equation with the indicated index convention. Reversing the wedge order changes the sign.
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