# Notation about basis of gamma matrices in $4d$

In Quantum Field theories, we encounter gamma matrices a lot.
Reading from various textbook, i encountered some textbook use different basis for their gamma matrices.

Gamma matrices are defined such that $\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\eta^{ab}$. Multiplying them in all possible way furnish the following list \begin{align} \{ \Gamma^A \} = \{1, \gamma^{a_1}, \gamma^{a_1 a_2}, \cdots \gamma^{a_1 \cdots a_d} \} \end{align} with $a_{1}<a_{2}<a_{3}\cdots<a_{d}$. where $d$ is the dimension of spacetime for given gamma matrices.

Applying above it for $4d$ i have \begin{align} \{ \Gamma^A \} = \{ 1, \gamma^{a_1}, \gamma^{a_1 a_2}, \gamma^{a_1 a_2 a_3}, \gamma^{a_1 a_2 a_3 a_4} \} \end{align}

In usual qft textbook, writes \begin{align} \{ \Gamma^A \} = \{1, \gamma_5, \gamma^{a_1}, \gamma_5 \gamma_{a_1}, \gamma_{a_1 a_2} \} \end{align}

I know they are equivalent, $i.e$, \begin{align} &\gamma_5 \propto \gamma^1 \gamma^2 \gamma^3\gamma^4 \propto \textrm{four product of gamma}\\ & \gamma_5 \gamma_{a_1} \propto \textrm{three products of gamma} \end{align}

What i am interested is instead of writing the first one modern qft textbook prefers to write the second form. Is there any reason for that? I think it might be just a matter of convention, like eastern or western approach of metric $(-1, 1, 1, \cdots, 1)$, $(1, -1, -1, \cdots -1)$, etc

In 4D Minkowski space the $\Gamma^{a}$'s has standard form. As $\Gamma^{a}=\mathbf{1}_{4\times4},\gamma^{\mu},\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}],\gamma^{5}\gamma^{\mu},\gamma^{5}$, altogether 16 matrices. Provided that $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\mathbf{1}_{4\times4}\eta^{\mu\nu}$.
When we construct bilinears with $\Gamma^a$s, the modern qft text book $\Gamma$s turn out to be useful. For a spin-$1/2$ Fermionic field $\psi$, $\bar{\psi}\psi$ transforms as a scalar, $\bar{\psi}\gamma^a\psi$ as a vector, $\bar{\psi}\gamma_5\psi$ as a pseudo scalar, $\bar{\psi}\gamma_5\gamma^a\psi$ as a pseudo vector, $\bar{\psi}\gamma^{a_1 a_2}\psi$ as a second-rank tensor under Lorentz transformations.