The next step is to compute the matrix elements of the operators J 1, J 2, J 3 and K 1, K 2, K 3 forming the basis of the Lie algebra of s o ( 3 ; 1 ) . The matrix elements of J ± ± = J 1 ± ± i J 2 and J 3 (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by ⟨ j m | J + | j m − − 1 ⟩ = ⟨ j m − − 1 | J − − | j m ⟩ = ( j + m ) ( j − − m + 1 ) , ⟨ j m | J 3 | j m ⟩ = m , where the labels j 0 and j 1 have been dropped since they are the same for all basis vectors in the representation.