\mbox{\tt potential angular momentum} \nonumber \\
 &=&
 \int \,\psi^\dagger \,\bm{x} \times 
 (\bm{p} - g \,\bm{A} ) \,
 \psi \,d^3 x \ + \ g \,\int \,\psi^\dagger \,
 \bm{x} \times \bm{A}_{phys} \,\psi \,d^3 x \nonumber \\
 &=&
 \int \,\psi^\dagger \,\bm{x} \times 
 (\bm{p} - g \,\bm{A}_{pure} ) \,
 \psi \,d^3 x \ \ \ \ \ \ = \ \ \ 
 \bm{L}^{\prime q} .\end{aligned}$$ Next, in the 2nd paper [@Wakamatsu11A], we found that we can make a covariant extension of gauge-invariant decomposition of the nucleon spin. Covariant generalization of the decomposition has several advantages.

-   First, it is useful to find relations to deep-inelastic-scattering (DIS) observables.

-   Second, it is vital to prove frame-independence of the decomposition.

-   Third, it generalizes and unifies the previously-known nucleon spin decompositions.

Basically, we find two physically different decompositions. The decomposition (I) contains the well-known Ji decomposition [@Ji97PRL], although it also allows gauge-invariant decomposition of gluon total angular momentum into its spin and OAM parts. The decomposition (II) contains in it three known decomposition, i.e. those of Bashinsky-Jaffe [@BJ99], of Chen et al. [@Chen08],[@Chen09], and of Jaffe-Manohar [@JM90], as we shall discuss below. The basis of our treatment is a decomposition of the full gauge field 