Lorentzian connection-form antisymmetry (${\omega }_{ab}=-{\omega }_{ba},{\omega }_{ab}={\eta }_{ac}{\omega }^c{}_b$)

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.