 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 ${\epsilon }_{ijk}$ is the Levi-Civita symbol.

Every angular momentum operator satisfies

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. ^[ref]