Showing that a given matrix provides the desired Lorentz transformation

I am working on problem 5.12 in Halzen and Martin’s Quarks and Leptons. We are asked to show that the matrix

$$S = 1 - \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu}$$

represents the infinitesimal Lorentz transformation

$$\Lambda^\mu {}_\nu = \delta^\mu {}_\nu + \varepsilon^\mu {}_\nu ,$$

in the sense that

$$S^{-1} \gamma^\mu S = \Lambda^\mu {}_\nu \gamma^\nu .$$

I have already proven that $S^{-1} = \gamma^0 S^\dagger \gamma^0$, so I can begin with the left-hand side of this third equation:

\begin{align} S^{-1} \gamma^\mu S &= \gamma^0 S^\dagger \gamma^0 \gamma^\mu S \\ &= \gamma^0 \left( 1 + \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \right) \gamma^0 \gamma^\mu \left( 1 - \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \right) \\ &= \gamma^0 \gamma^0 \gamma^\mu - \gamma^0 \gamma^0 \gamma^\mu \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} + \gamma^0 \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \gamma^0 \gamma^\mu - \gamma^0 \frac{i}{4} \sigma_{\rho \tau} \varepsilon^{\rho \tau} \gamma^0 \gamma^\mu \frac{i}{4} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \\ &= \gamma^\mu + \frac{1}{16} \gamma^0 \sigma_{\rho \tau} \varepsilon^{\rho \tau} \gamma^0 \gamma^\mu \sigma_{\mu \nu} \varepsilon^{\mu \nu} \\ &= \gamma^\mu + \frac{1}{16} \left( \sigma_{\rho \tau} \varepsilon^{\rho \tau} \sigma_{\mu \nu} \varepsilon^{\mu \nu} \right) \gamma^\mu . \end{align}

Remember, we want this to be equal to

$$\Lambda^\mu {}_\nu \gamma^\nu = \left( \delta^\mu {}_\nu + \varepsilon^\mu {}_\nu \right) \gamma^\nu = \gamma^\mu + \varepsilon^\mu {}_\nu \gamma^\nu .$$

The first term is there already, but I have no idea how that second term is going to work out to $\varepsilon^\mu {}_\nu \gamma^\nu$. Can anyone give me a hint, or tell me what I’ve done wrong?

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To order $\epsilon$, isn't $S^{-1}$ just $1+\frac{i}{4}\sigma_{\mu\nu}\epsilon^{\mu\nu}$ ? Then try expanding out $S^{-1}\gamma^{\mu}S$ keeping terms of order $\epsilon$ and using the Clifford algebra relations when you have to commute $\gamma$s through $\sigma_{\mu\nu}$ terms...