CSCO of Total Angular momentum

[QuantumDuality]

I understand that in a system composed of two articles, the total angular momentum is:
J = J₁ + J₂
From the operators: J^2, J_z, J_1z, J_2z，J^2_1z，J^2_2z,
I get two possible sets of operators that commute:
{J^2, J_z, J^2_1z, J^2_2z} and {J^2_1z, J^2_2z, J_1z, J_2z}
What I don't understand is why the second set is the CSCO

 

[DrClaude]

Because the Hamiltonian includes $\hat{J}^2$. You always have to take $\hat{H}$ as part of the CSCO.