 Solution

Since $S^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$, apply the commutator derivation identity to this commutator and insert part (ii):This is exactly the printed Lorentz algebra relation, including its final two signs; their alternative appearance in other formulas comes from $S^{\rho \mu }=-S^{\mu \rho }$. Therefore the matrices S form a representation of the Lorentz algebra. Their definition contains no extra factor of $i$, so the spatial rotation generators below are represented by skew-Hermitian matrices. Adding a factor of $i$ would change the convention for the Lie algebra structure constants and cannot be done silently.