From this question https://physics.stackexchange.com/questions/101887/do-gamma-matrices-form-a-basis, I know that the gamma matrices can be used as the generators of an algebra with 16 basis elements as follows:
- 1 the identity matrix $\mathbb{1}$
- 4 matrices $\gamma^\mu$
- 6 matrices $\sigma^{\mu\nu}=\gamma^{[\mu}\gamma^{\nu]}$
- 4 matrices $\sigma^{\mu\nu\rho}=\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}$
- 1 matrix $\sigma^{\mu\nu\rho\delta}=\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\delta]}=i\epsilon^{\mu\nu\rho\delta}\gamma^5$
In the answers, it is claimed that linear combination with complex coefficients of these 16 elements forms a complete basis of $M(4,\mathbb{C})$.
My question is: Is a linear combination with real coefficient forms a complete basis of $M(4,\mathbb{R})$?