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
 A: I'd argue that the root of this is that different Gamma matrix bases give you different "good" features, and while the choice is equivalent (and amounts to a choice of basis for your Dirac spinor), which choice is "right" depends on what features you want to be obvious/trivial -- particle/antiparticle, left/right handedness, ease of performing Legendre transforms, etc.  Which of these features you want to expose depends on what type of research you want to do, so different textbooks will likely reflect choices that optimize study in the hot research subfield at the time of writing.
Ultimately, however, the only thing that is physical is that the Gamma matrices obey the Clifford algebra.  
A: 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 $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\mathbf{1}_{4\times4}\eta^{\mu\nu}$. 
A: 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.
From these bilinears, one can construct interaction terms which give distinct features to the S-Matrix. The V-A interaction model for eg. had vector and axial(peudo) vector currents.
