# Fierz like identity for $\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}$

It is known that contracting over the vector indices of two Pauli matrices (the 3d γ-matrices) can be simplified to a bunch of δ functions. This is done via the Fierz formula $$\delta_{ab}\sigma^a_{ij}\sigma^b_{kl}=\delta_{ij}\delta_{kl}-2\epsilon_{ik}\epsilon_{jl}=2\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl}.$$ Here, $a,b,c=1,2,3$ are vector indices, and $i,j,k,l=1,2$ are spinor indices. Repeated indices are summed over.

However, I would like to know whether there is a similar formula for $$\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}~?$$ If so, what is it?

$$\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}= 2i \Big(\delta ^i_q \delta ^k_j\delta ^p_l - \delta ^i_l\delta ^k_q\delta ^p_j\Big ),$$ with all the right pairwise interchange symmetries. (I have raised, for convenience, not terribly meaningfully, indices in the Kronecker δs to churn these interchanges more easily.)
The normalization ensues by taking the trace, $\delta ^i_i\delta ^k_k\delta ^p_p -\delta ^i_i=6$, given the SU(2) Lie algebra normalization/identity, $${\mathrm Tr} ~ \sigma^a \sigma^b \sigma^c =2i~\epsilon_{abc} ~.$$