Define the spinor generators
$$
S^{\mu\nu}=\frac14[\gamma^\mu,\gamma^\nu].
$$
Repeated use of the <Clifford algebra> gives
$$
\begin{aligned}
[S^{\mu\nu},\gamma^\rho]
&=\frac14(\gamma^\mu\gamma^\nu\gamma^\rho
-\gamma^\nu\gamma^\mu\gamma^\rho
-\gamma^\rho\gamma^\mu\gamma^\nu
+\gamma^\rho\gamma^\nu\gamma^\mu)\\
&=\gamma^\mu\eta^{\nu\rho}-\gamma^\nu\eta^{\rho\mu}.
\end{aligned}
$$
The <Spinor representation of the Lorentz group> is
$$
S[\Lambda]=\exp\left(\frac12\Omega_{\rho\sigma}S^{\rho\sigma}\right).
$$
The adjoint relation for the Dirac matrices implies
$$
(S^{\rho\sigma})^\dagger=-\gamma^0S^{\rho\sigma}\gamma^0.
$$
Exponentiating yields
$$
\boxed{S[\Lambda]^\dagger=\gamma^0S[\Lambda]^{-1}\gamma^0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 3 / i / Solution

{c}

With $S^{\mu\nu}=\frac14[\gamma^\mu,\gamma^\nu]$, the spinor representation is
$$
S[\Lambda]=\exp\left(\frac12\Omega_{\mu\nu}S^{\mu\nu}\right).
$$
It obeys $S^{-1}\gamma^\mu S=\Lambda^\mu{}_\nu\gamma^\nu$.


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 Incoming links: Spinor representation of the Lorentz group