Transpose symmetry of a charge-conjugation matrix
= Transpose symmetry of a charge-conjugation matrix
{title2=$C^T=\pm C$}
If $C\gamma^{\mu T}C^{-1}=-\gamma^\mu$, transposition shows that $C^TC^{-1}$ commutes with every <gamma matrix>. Irreducibility of the complex Dirac representation and the <Schur lemma> make it scalar. Transposing $C^T=zC$ again yields $z^2=1$, so $C^T=\pm C$.