Proof of two Lorentz-algebra identities

Question

I am currently working through the QFT introduction text by Peskin and Schroeder and try to fill in two identities that I wasn't able to prove (it should be fairly simple, but my experience with this kind of calculations is limited).

1. $$ [\gamma^{\mu}, S^{\rho\sigma}] = (\mathcal{J}^{\rho\sigma}~)^{\mu}{}_{\nu}\gamma^{\nu}$$ where $\gamma^{\mu}$ are the gamma-matrices in Weyl-representation (don't think the specific representation matters though), $S^{\rho\sigma} = \frac{i}{4}[\gamma^{\rho}, \gamma^{\sigma}~]$ and $(\mathcal{J}^{\rho\sigma}~)_{\mu\nu} = i(\delta^{\rho}{}_{\mu}~\delta^{\sigma}{}_{\nu} - \delta^{\rho}{}_{\nu}~\delta^{\sigma}{}_{\mu})$ a specific representation of the Lorentz algebra.
2. $$\left(1+\frac{i}{2}\omega_{\rho\sigma}~S^{\rho\sigma}\right)\gamma^\mu\left(1-\frac{i}{2}\omega_{\rho\sigma}~S^{\rho\sigma}\right) = \left(1-\frac{i}{2}\omega_{\rho\sigma}~\mathcal{J}^{\rho\sigma}\right)^{\mu}{}_{\nu}~\gamma^{\nu}$$ using the result of 1. I especially do not understand where the term $$ \dfrac{1}{4}\omega_{\rho\sigma}~S^{\rho\sigma}\gamma^{\mu}\omega_{\rho\sigma}~S^{\rho\sigma}$$ is going to?

Answer 1

Since I finally got the first one right, I might as well answer my own question.

\begin{align*} [\gamma^{\mu}, S^{\rho\sigma}] &= \dfrac{i}{4}[\gamma^{\mu}, [\gamma^{\rho}, \gamma^{\sigma}]] \\ &= \dfrac{i}{4}\left( \gamma^{\mu}\gamma^{\rho}\gamma^{\sigma} - \gamma^{\mu}\gamma^{\sigma}\gamma^{\rho} - \gamma^{\rho}\gamma^{\sigma}\gamma^{\mu} + \gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\right) \\ &= \dfrac{i}{4}\left( [2g^{\mu\rho} - \gamma^{\rho}\gamma^{\mu}]\gamma^{\sigma} - [2g^{\mu\sigma} - \gamma^{\sigma}\gamma^{\mu}]\gamma^{\rho} - \gamma^{\rho}\gamma^{\sigma}\gamma^{\mu} + \gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\right) \\ &= \dfrac{i}{4}\left(-\gamma^{\rho}[2g^{\mu\sigma}-\gamma^{\sigma}\gamma^{\mu}] + \gamma^{\sigma}[2g^{\mu\rho}-\gamma^{\rho}\gamma^{\mu}] - \gamma^{\rho}\gamma^{\sigma}\gamma^{\mu} + \gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\right) + \dfrac{i}{2}\left( g^{\mu\rho}\gamma^{\sigma} - g^{\mu\sigma}\gamma^{\rho}\right)\\ &= i\left( g^{\mu\rho}\gamma^{\sigma} - g^{\mu\sigma}\gamma^{\rho}\right) \\ &= i \left( \delta^{\rho}_{~\mu}\delta^{\sigma}_{~\nu} -\delta^{\sigma}_{~\mu}\delta^{\rho}_{~\nu} \right)g^{\mu\nu}\gamma^{\nu} \\ &= \left( \mathcal{J}^{\sigma\rho} \right)^{\mu}_{~\nu}\gamma^{\nu} \end{align*}

When taking into consideration that $\omega$ is infinitesimal the second identity is trivial. Neglecting higher order terms in $\omega$ and using $(1)$ yields \begin{align*} \left(1+\frac{i}{2}\omega_{\rho\sigma}~S^{\rho\sigma}\right)\gamma^\mu\left(1-\frac{i}{2}\omega_{\rho\sigma}~S^{\rho\sigma}\right) &= \gamma^\mu + \frac{i}{2}\omega_{\rho\sigma}[S^{\rho\sigma}, \gamma^\mu]\\ &= \left(1-\frac{i}{2}\omega_{\rho\sigma}~\mathcal{J}^{\rho\sigma}\right)^{\mu}{}_{\nu}~\gamma^{\nu} \end{align*}