Consider the generators of $SO(n)$, written as $M_{\mu\nu} = - M_{\nu\mu}$ and they satisfy $$ \left[ M_{\mu\nu} , M_{\rho\sigma} \right] = i \left( \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\mu\sigma} M_{\nu\rho} - \eta_{\mu\rho} M_{\nu\sigma} - \eta_{\nu\sigma} M_{\mu\rho} \right) $$ Now, consider the spinor representation $$ \Sigma_{\mu\nu} = \frac{i}{4} \left[ \gamma_\mu , \gamma_\nu \right] $$ where $\gamma_\mu$ are the Dirac matrices that satisfy $\{ \gamma_\mu , \gamma_\nu \} = 2 \delta_{\mu\nu}$.
Is there a general formula for the quantity $$ \left( \Sigma_{\mu\nu} \right)_{ab} \left( \Sigma^{\mu\nu} \right)_{cd} $$ for arbitrary $a,b,c,d$?
If so, what is the same quantity for $SO(n-1,1)$ instead?
Thanks!
Yes, this is one of the celebrated Fierz identities, and in particular the middle row of Kennedy 1982 last matrix on p 1335; something like $$ \left( \Sigma_{\mu\nu} \right)_{ab} \left( \Sigma^{\mu\nu} \right)_{cd} = -\frac{1}{2} \left( \Sigma_{\mu\nu} \right)_{ad} \left( \Sigma^{\mu\nu} \right)_{cb} -\frac{3}{2}\delta_{ad} \delta_{cb} -\frac{3}{2} \gamma^5_{ad}\gamma^5_{cb}, $$ where absolutely no attempt has been made to align the normalizations of the gamma basis to your idiosyncratic one, and the WP conventions have been used.
The prudent thing to do is to throw in plug-in entries or take suitable trace projections, to clean up each and every one of them, in your conventions.
Related. Also eqn(236) Good 1955; eqn (5) De Vries & Van Zanten 1970. For arbitrary SO(N,1), see Kennedy 1981.