= Cartan curvature equation with input indices
{c}
{title2=$\Theta=d\theta-\theta\wedge\theta$}
Write $De_i=\sum_j\theta_{ij}\otimes e_j$, with the differentiated input index first. The graded <Leibniz rule> gives $D^2e_i=\sum_j(d\theta_{ij}-\sum_k\theta_{ik}\wedge\theta_{kj})\otimes e_j$. For frames $e'=ge$, the <connection matrix> transforms as $\theta'=dg\,g^{-1}+g\theta g^{-1}$ and curvature by $\Theta'=g\Theta g^{-1}$. Transposing to the usual coefficient-column convention gives $A=\theta^t$ and $F=dA+A\wedge A=\Theta^t$; the opposite sign is an index convention, not a different curvature.
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