= Solution
The <commutator> with the <scalar> metric term vanishes. Reorder the third <gamma matrix> using the <Clifford algebra> twice:
$$
\gamma^\mu\gamma^\nu\gamma^\rho
=2\eta^{\nu\rho}\gamma^\mu-2\eta^{\mu\rho}\gamma^\nu+\gamma^\rho\gamma^\mu\gamma^\nu.
$$
Subtract the last term and multiply by $1/2$ to obtain
$$
\boxed{[S^{\mu\nu},\gamma^\rho]=\eta^{\nu\rho}\gamma^\mu-\eta^{\mu\rho}\gamma^\nu,\qquad C=1,\ D=-1.}
$$
This is the infinitesimal <vector> transformation law implemented by the <Spinor representation of the Lorentz group>. The first contraction is $\eta^{\nu\rho}$ as printed in the PDF; the TeX conversion's $\eta^{\rho\rho}$ is erroneous and would fail, for example, when $\mu=0,\nu=1,\rho=2$: the actual <commutator> is zero.
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