= Lorentzian connection-form antisymmetry
{c}
{title2=$\omega_{ab}=-\omega_{ba},\quad\omega_{ab}=\eta_{ac}\omega^c{}_b$}
In an <orthonormal coframe in spacetime>, antisymmetry applies after lowering the first index with the Lorentzian frame metric. Thus mixed time–space <connection 1-forms> are equal under index interchange, while mixed spatial forms are opposite. The same rule holds for <curvature 2-forms>. Confusing lowered antisymmetry with ordinary mixed-index antisymmetry changes the curvature signs.
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