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By separating the gluon field into physical and pure-gauge components, theusual Poincar\'e subalgebra for an interacting system can be reconciled withgauge-invariance when decomposing the total rotation and translation generatorsof QCD into quark and gluon parts. The gauge-invariant quark/gluon parts act asthe generators for the gauge-invariant physical component of the quark/gluonfield, not the full quark/gluon field which also contains the gauge degrees offreedom. We clarify that the naive canonical decomposition of generators, whiletrivially respecting the Poincar\'e subalgebra, might not give a completelygauge-invariant quark-gluon structure of the nucleon momentum and spin, thoughlimited invariance within a certain gauge class can be proven.

Poincar\'e subalgebra and gauge invariance in nucleon structure