= Solution
Since $S^{\rho\sigma}=\frac14[\gamma^\rho,\gamma^\sigma]$, apply the <commutator derivation identity> to this <commutator> and insert part (ii):
$$
\begin{aligned}
[S^{\mu\nu},S^{\rho\sigma}]
&=\frac14\left([[S^{\mu\nu},\gamma^\rho],\gamma^\sigma]+[\gamma^\rho,[S^{\mu\nu},\gamma^\sigma]]\right)\\
&=\eta^{\nu\rho}S^{\mu\sigma}-\eta^{\mu\rho}S^{\nu\sigma}+\eta^{\nu\sigma}S^{\rho\mu}-\eta^{\mu\sigma}S^{\rho\nu}.
\end{aligned}
$$
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 \b[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.
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