QCD, which means that each term is separately gauge-invariant [@Wakamatsu11A]. These conditions are general enough, so that they are expected to be satisfied by most gauges used in QCD. The fact that the Bashinsky-Jaffe decomposition is contained in our more general decomposition (II) was explicitly verified in [@Wakamatsu11A]. It is also logically plausible that the Chen el al. decomposition is contained in our decomposition (II). This is because, although their decomposition is given in a noncovariant manner like the formulation of electrodynamics to be discussed in sec.II, their decomposition of the gluon field $\bm{A}$ into $\bm{A}_{phys}$ and $\bm{A}_{pure}$ naturally satisfy our general conditions (\[gcond1\]),(\[gcond2\]),(\[gcond3\]). In view of the fact that both frameworks of Bashinsky-Jaffe and of Chen et al. are contained in our more general gauge-invariant decomposition, we naturally expect that both give the same answer at least for the momentum sum rule of QCD as well as for the longitudinal spin decomposition of the nucleon, which can be formulated frame-independently.

In a recent paper, Ji, Xu, and Zhao threw doubt on this viewpoint [@JXZ12]. According to them, the Bashinsky-Jaffe decomposition is one gauge-invariant extension (GIE) of gauge-variant Jaffe-Manohar decomposition based on the light-cone gauge, whereas the Chen el al’s 