spin operators. This problem was attacked and solved in the paper [@Wakamatsu11B]. We find that the calculation in the Feynman gauge (as well as in any covariant gauge including the Landau gauge) reproduces the answer obtained in the LC gauge, which is also the answer obtained in the famous Altarelli-Parisi method [@AP77]. (This conclusion for the evolution of $\Delta G$ however contradicts the one given in [@Chen11B].) Our finding is important also from another context. So far, a direct check of the answer of Altarelli-Pasiri method for the evolution of $\Delta G$ within the operator-product-expansion (OPE) framework was limited to the LC gauge, because it was believed that there is no gauge-invariant definition of gluon spin in the OPE framework. This was the reason why the question of gauge-invariance of $\Delta G$ has been left in unclear status for a long time.

After establishing satisfactory natures of the decomposition (I), now we turn our attention to another decomposition (II). According to Chen et al., the greatest advantage of the decomposition (II) is that their quark OAM operator $\bm{L}^\prime_q$ satisfies the standard commutation relation of angular momentum : $$\bm{L}^\prime_q \times \bm{L}^\prime_q \ = \ i \,\bm{L}^\prime_q ,$$ due to the property 