= Angular momentum commutation relations
The <angular momentum commutation relations> express how generators of spatial <rotations> fail to commute. Here $J^2=J_x^2+J_y^2+J_z^2$ and $\varepsilon_{ijk}$ is the <Levi-Civita symbol>.
Every <angular momentum> operator satisfies
$$
[J_i,J_j]=i\hbar\varepsilon_{ijk}J_k,
\qquad
[J^2,J_i]=0.
$$
Since $J^2$ commutes with every component, states can be labeled simultaneously by total <angular momentum> and one chosen projection. Distinct components need not share a complete <eigenbasis>. The same algebra governs both <orbital angular momentum> and intrinsic <spin>, although their allowed representations differ. <Quantum Mechanics by David Tong>[Section 4.1.1, angular momentum commutators and simultaneous eigenstates]{ref}
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