Lorentz-spinor generators from a Clifford algebra ($M^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$)

In a real-generator convention, $M^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$ obeys $[M^{\rho \sigma },{\gamma }^{\mu }]=g^{\sigma \mu }{\gamma }^{\rho }-g^{\rho \mu }{\gamma }^{\sigma }$. Applying this identity to the commutator of two gamma matrices proves the Lorentz algebra relations. With the matching vector generators and real antisymmetric parameters, $S=\exp (\frac{1}{2}{\Omega }_{\rho \sigma }{M}^{\rho \sigma })$ obeys $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. A convention using Hermitian Lorentz generators instead must consistently insert factors of $i$.