\ = \ M^{\prime \mu \nu \lambda}_{q-spin} \ + \ 
 M^{\prime \mu \nu \lambda}_{q-OAM} \ + \ M^{\prime \mu \nu \lambda}_{G-spin}
 \ + \ M^{\prime \mu \nu \lambda}_{G-OAM} \ + \ M^{\prime \mu \nu \lambda}_{boost},
 \label{decompositionII}$$ with $$\begin{aligned}
 M^{\prime \mu \nu \lambda}_{q-spin}
 &=& M^{\mu \nu \lambda}_{q-spin}, \\
 M^{\prime \mu \nu \lambda}_{q-OAM}
 &=& \bar{\psi} \,\gamma^\mu \,(\,x^{\nu} \,i \,D^{\lambda}_{pure} 
 \ - \ x^{\lambda} \,i \,D^{\nu}_{pure} \,) \,\psi \label{decomposition2B} \\
 M^{\prime \mu \nu \lambda}_{G-spin} 
 &=& M^{\mu \nu \lambda}_{G-spin}, \\
 M^{\prime \mu \nu \lambda}_{G-OAM} 
 &=& - \,2 \,\mbox{Tr} \,[\, F^{\mu \alpha} \,
 (\,x^{\nu} \,D^{\lambda}_{pure}
 \ - \ x^{\lambda} \,D^{\nu}_{pure} \,) \,A_{\alpha}^{phys} \,] ,
 \label{decomposition2E}\end{aligned}$$ and $$M^{\prime \mu \nu \lambda}_{boost} 
 \ = \ - \,\frac{1}{2} \,\mbox{Tr} \,F^2 \,(\,x^{\nu} \,g^{\mu \lambda} 
 \ - \ x^{\lambda} \,g^{\mu \nu} \,) .$$ At first sight, this decomposition looks like a covariant generalization of Chen et al’s decomposition, in the sense that the quark OAM part contains [*pure gauge covariant derivative*]{}. However, a crucial difference is that we have not yet fixed the gauge (and the Lorentz frame) explicitly. The point is that, as long as the general conditions (\[gcond1\]), (\[gcond2\]), (\[gcond3\]) are satisfied, each term of the decomposition (II) is never mixed up under general color gauge transformation of 