Angular momentum addition

Coupling two quantum angular momenta $j_1,j_2$ into total $j=|j_1-j_2|,\dots ,j_1+j_2$. Scalar products follow from $2{\mathbf{J}}_1\cdot {\mathbf{J}}_2={\mathbf{J}}^2-{\mathbf{J}}_1^2-{\mathbf{J}}_2^2$.