Does the following, and similar identities (in 4 dimensions) depend on the normalization of gamma matrices? $$ \gamma^a \gamma^b \gamma^c=g^{ab}\gamma^c+g^{bc}\gamma^a-g^{ac}\gamma^b-i\epsilon^{dabc}\gamma^d\gamma^5~. $$

What I'm asking is, with $\{\gamma^a,\gamma^b\}=2g^{ab}$, gamma matrices in $(-,+,+,+)$ and $(+,-,-,-)$ metric conventions differ by a factor of $i$. So, naively multiplying all the gamma matrices above by $i$ leads to a sign reversal on either of sides. Would it be correct?



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