Proper quark-gluon vertex colour factors ($C_F-C_A/2,C_A/2$)

For Hermitian fundamental representation generators normalized by $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, the Abelian-shaped one-loop Feynman diagram has colour factor ${\sum }_b{T}^b{T}^a{T}^b=(C_F-C_A/2)T^a=-T^a/(2N)$. Expand ${\sum }_b[T^b,[T^b,T^a]]=C_A{T}^a$ to prove the identity. With oriented structure tensor $f^{abc}$, the three-gluon graph has $f^{abc}{T}^b{T}^c=i{C}_A{T}^a/2$ 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 $3/2$.