Loday in the article available here http://www.numdam.org/item/?id=ASENS_1976_4_9_3_309_0, described the Quillen +-construction in the proof of the Theorem 1.1.1. I don't understand his last equality, $j \circ ∂(e_{\alpha}^{3}) = j \circ \rho ([b_{\alpha}])$.
Does anyone see why this is true?