next section, by utilizing a plainer example from electrodynamics.) One is an extension of the Ji decomposition, which also fulfills the decomposition of the gluon total angular momentum into the intrinsic spin and orbital part, while the other is a decomposition that contains in it three known decompositions as [*gauge-fixed forms*]{} of more general expression. (We however recall that there a criticism to this idea [@JXZ12].) The orbital OAMs appearing in these two decompositions are respectively the [*dynamical*]{} OAMs and the [*generalized canonical*]{} OAMs. Since both decompositions are gauge-invariant, there arises a possibility that they both correspond to observables.

A clear relation with observables was first obtained for the decomposition (I) [@Wakamatsu11A]. The keys are the following identities, which hold in our decomposition (I). For the quark part, it holds that $$x^\nu \,T^{\mu \lambda}_q \ - \ x^\lambda \,T^{\mu \nu}_q \ = \ 
 M^{\mu \nu \lambda}_{q-spin} \ + \ M^{\mu \nu \lambda}_{q-OAM} \ + \
 \mbox{total divergence} ,$$ while for the gluon part we have $$x^\nu \,T^{\mu \lambda}_g \ - \ x^\lambda \,T^{\mu \nu}_q \ - \ 
 \mbox{boost} \ = \ 
 M^{\mu \nu \lambda}_{g-spin} \ + \ M^{\mu \nu \lambda}_{g-OAM} \ + \
 \mbox{total divergence}.$$ Here, $T^{\mu \nu}_q$ and $T^{\mu 