Lorentz generator from gamma-matrix commutators ($S^{\mu \nu }=\frac{1}{4}[{\gamma }^{\mu },{\gamma }^{\nu }]$)

The Clifford algebra gives $S^{\mu \nu }=\frac{1}{2}{\gamma }^{\mu }{\gamma }^{\nu }-\frac{1}{2}{\eta }^{\mu \nu }I$. Reordering one more gamma matrix gives $[S^{\mu \nu },{\gamma }^{\rho }]={\eta }^{\nu \rho }{\gamma }^{\mu }-{\eta }^{\mu \rho }{\gamma }^{\nu }$. The commutator derivation identity then givesso these matrices represent the Lorentz algebra. This convention has no extra factor of $i$; the spatial generators are $-\frac{i}{2}{ϵ}^{jkl}\mathrm{diag}({\sigma }^l,{\sigma }^l)$.