= Antisymmetry of the four-dimensional charge-conjugation matrix
{title2=$C^T=-C$}
In the <Weyl representation of the gamma matrices>, $C=i\gamma^2\gamma^0=\operatorname{diag}(\varepsilon,-\varepsilon)$ with $\varepsilon=i\tau_2$. The antisymmetry of $\varepsilon$ gives $C^T=-C$, and direct multiplication gives $C\gamma^{\mu T}C^{-1}=-\gamma^\mu$ and $C^\dagger C=I$. Any two such intertwining matrices differ by a scalar: their ratio commutes with all <gamma matrices>, whose products span the full four-by-four matrix algebra. Changing spinor basis sends $C$ to $SCS^T$, so transpose antisymmetry is basis-independent. Its sign is therefore fixed, not merely one of the two alternatives supplied by the general <transpose symmetry of a charge-conjugation matrix> argument.
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