Calculating the quark propagator while dealing with the color, Dirac & flavor matrices all together

$S^+$ is a diagonal matrix in color space, and is given as : $S^+ = S_{rg} ^+ hat{P_{rg}} + S_b ^+ hat{P_b}, where, hat{P_b} = 1_c – hat{P_{rg}}$
is the projector on blue quarks, $hat{P_{rg}}$ projects onto the "red" and "green" color components, and, $S_b ^+ = (require{cancel}{p_+})^{-1} = frac{cancel{p_+}}{p_+ ^2}$.
Now, $T^+$ is a component of the Nambu-Gorkov propagator given as $T^+ = frac{triangle ^* gamma _5 tau _2 lambda _2 (cancel{p_-}cancel{p_+})}{D(p)} – frac{triangle ^* gamma _5 tau _2 lambda _2 |triangle|^2}{D(p)} rightarrow T^+ = bigg[(-) frac{(cancel{p_-}cancel{p_+})triangle ^* gamma _5 tau _2 lambda _2}{D(p)} -(-) frac{|triangle|^2 triangle ^* gamma _5 tau _2 lambda _2}{D(p)}bigg]; $
$overrightarrow{tau} and lambda$‘s are the Pauli and Gell-Mann matrices respectively, $triangle ^* and triangle$ are scalars, $cancel{p_pm}equivcancel{p} pm mu gamma ^0$. Question: Are the extra negative signs (shown in round brackets) in the last expression for
$T^+$ due to the anticommuting nature of $(triangle ^* gamma _5 tau _2 lambda _2)$ with $(cancel{p_-}cancel{p_+})$, or, beacuse $(cancel{p_-}cancel{p_+})$ ‘lives’ in color space while "$triangle ^* gamma _5 tau _2 lambda _2$" do not (but, as per the definitions of $cancel{p_pm}$ which include the Dirac matrix $gamma _0$, shouldn’t it ‘be living’ in Dirac space)?
For further refernce, [please refer to Problem 4 in this paper].1