= Local Lorentz transformation
{title2=$e'^A{}_\mu=\Lambda^A{}_B(x)e^B{}_\mu$}
An <orthonormal coframe in spacetime> can be rotated by a position-dependent Lorentz matrix without changing the spacetime <metric tensor>. Its <spin connection> transforms as $\omega'=\Lambda\omega\Lambda^{-1}-d\Lambda\,\Lambda^{-1}$. A <spinor field> transforms through a chosen spin lift $\psi'=S(\Lambda)\psi$, not through the vector matrix $\Lambda$. With $S^{-1}\gamma^A S=\Lambda^A{}_B\gamma^B$, an infinitesimal parameter gives $S=1+\tfrac14\alpha_{AB}\gamma^{AB}+O(\alpha^2)$, where $\gamma^{AB}=\tfrac12[\gamma^A,\gamma^B]$. Consistent global transition lifts require a <spin structure>; a $2\pi$ spatial rotation lifts to minus the identity on spinors.
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