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A complete set of commuting observables consists of mutually commuting quantum observables whose joint eigenvalues distinguish all vectors of a simultaneous eigenbasis, up to phase. For a finite-dimensional Hilbert space, commuting self-adjoint operators admit a simultaneous orthonormal basis of eigenvectors; completeness means that every joint eigenspace is one-dimensional. An additional quantum observable that is already a function of the others supplies no independent quantum number.

For the addition of angular momentum, the uncoupled labels $(j_1,j_2,m_1,m_2)$ correspond to $\{J_1^2,J_2^2,J_{1z},J_{2z}\}$, whereas the coupled labels $(j_1,j_2,j,m)$ correspond to $\{J_1^2,J_2^2,J^2,J_z\}$. Within a fixed pair of irreducible angular momentum spaces each choice gives a natural eigenbasis. The Clebsch-Gordan coefficients implement the change between them.

The operators associated with the two angular momenta commute with one another, and each squared angular momentum operator commutes with every component of that same angular momentum. Two useful identities areConsequently the two maximal commuting subsets of the displayed collection are

The first set gives the uncoupled eigenbasis $|j_1{m}_1⟩\otimes |j_2{m}_2⟩$. Its $J_z$ eigenvalue is already fixed by $m=m_1+m_2$. The second gives the coupled eigenbasis $|j_1{j}_2;jm⟩$ for the addition of angular momentum. Its scalar-product eigenvalue is already fixed by $j,j_1,j_2$. Thus an independent complete set of commuting observables uses four operators in either case; literal maximality within the question's list includes the dependent fifth operator.

For generic nonzero angular momenta, no further listed operator can be added. In particular,is generally nonzero, and $[J_{1z},J^2]=2[J_{1z},{\mathbf{J}}_1\cdot {\mathbf{J}}_2]$. The same obstruction applies to $J_{2z}$. Their sum $J_z$ commutes because the two commutators cancel.