= Proper quark-gluon vertex colour factors
{title2=$C_F-C_A/2,\ C_A/2$}
For Hermitian <fundamental representation> generators normalized by $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, the Abelian-shaped one-loop <Feynman diagram> has <colour factor> $\sum_bT^bT^aT^b=(C_F-C_A/2)T^a=-T^a/(2N)$. Expand $\sum_b[T^b,[T^b,T^a]]=C_AT^a$ to prove the identity. With oriented structure tensor $f^{abc}$, the three-gluon graph has $f^{abc}T^bT^c=iC_AT^a/2$ by antisymmetrizing the generator product. The factor $i$ and its sign must be combined consistently with the chosen three-gluon <Feynman rule>. For $SU(3)$, the two real coefficients are $-1/6$ and $3/2$.
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