# Gamma matrices and the metric

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.

• The components of $\gamma$ are matrices, and the components of the metric are a scalar, so I believe they should commute. – JamalS Mar 27 at 12:11
• But a matrix of matrices is still a matrix. Can you expound a bit? – Lopey Tall Mar 27 at 12:14
• 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. – JamalS Mar 27 at 12:16
• Ok I see my misunderstanding! Thank you! – Lopey Tall Mar 27 at 12:19