This is a slightly subjective question: Is there a "natural" way to justify the definition of $\gamma_5$? I have something like this in mind: Apart from Clifford algebra obeyed by the $\gamma$s I start with the conditions $$\{\gamma_5,\gamma^\mu\}=0$$ $$\gamma_5^2=1$$ I impose $(i)$ because I want something to commute with pairs of $\gamma_{\mu}$s so that $\gamma_5$ gives me notion of something which is Lorentz invariant (chirality) and I impose the involution condition i.e. $(ii)$ so that I have two subspaces to project to.
Can I arrive at $\gamma_5=i \gamma_0 \gamma_1 \gamma_2 \gamma_3$ uniquely somehow now? If not, what other conditions do I need to impose to "derive" this expression? Or is it hopelessly non-unique?
Probably a hint: In the last part of this answer the answerer recalls that a Schur lemma argument shows all such are in the span of $1$ and $\gamma_5$. This could be a way to show uniqueness of $\gamma_5$ once we have the expression, but I am unsure how the argument works. To my knowledge, Schur lemma implies irreducibility of the Dirac representation (see e.g. this and the user might be misremembering the usage or I can't see how it is related if at all it is.
Probably another hint: $\gamma_5$ can also be written as $$\gamma_5=\frac{i}{4!}\varepsilon^{\mu \nu \rho \sigma} \gamma_{\mu} \gamma_{\nu} \gamma_{\rho} \gamma_{\sigma}$$ The $\varepsilon$ appears in discussions of chirality/parity, etc. Is there a more natural way to motivate this definition than the other one?
