There is a step in a Dirac spinor manipulation problem I'm working on that requires me to make the following remark, $$ 2 \gamma^\sigma \eta^{\rho\mu}\mathbb{1}_4 \gamma^\nu = 2 \eta^{\rho\mu}\mathbb{1}_4 \gamma^\sigma \gamma^\nu $$ It is not clear to me how the gamma matrices commute with the metric... I know the gamma matrices anti-commute with one another but I don't see how the former is true.

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    $\begingroup$ The components of $\gamma$ are matrices, and the components of the metric are a scalar, so I believe they should commute. $\endgroup$ – JamalS Mar 27 at 12:11
  • $\begingroup$ But a matrix of matrices is still a matrix. Can you expound a bit? $\endgroup$ – Lopey Tall Mar 27 at 12:14
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    $\begingroup$ All the indices are free right? There are no contractions. So the left and right hand side must be true for all inputed values of the indices. So let's take $\gamma^{1} \eta^{00}$ for example. Since $\eta^{00} =1$ clearly any matrix will commute with a scalar. $\endgroup$ – JamalS Mar 27 at 12:16
  • $\begingroup$ Ok I see my misunderstanding! Thank you! $\endgroup$ – Lopey Tall Mar 27 at 12:19

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