SO(4) group ($SO(4)$)

The SO(4) group consists of real orthogonal four-by-four matrices of determinant one. It preserves orientation and the Euclidean metric. Its Spin(4) double cover identifies it with the quotient of two copies of SU(2) by their simultaneous central sign.

Tensor products $V_{j_L}\otimes V_{j_R}$ give the irreducible representations of the Spin(4) double cover. The simultaneous central sign acts by $(-1{)}^{2j_L+2j_R}$, so exactly the integer-sum pairs descend to SO(4). Their dimensions are $(2j_L+1)(2j_R+1)$. Under the diagonal spatial rotation subgroup, the Clebsch-Gordan decomposition for SU2 gives spins from $|j_L-j_R|$ through $j_L+j_R$ in steps of one.