Finite-dimensional complex Lorentz representations ($D^{(j_L,j_R)}(S)={\mathrm{Sym}}^{2j_L}S\otimes {\mathrm{Sym}}^{2j_R}\overline{S}$)

The complexified Lorentz algebra splits into two commuting ${\mathfrak{sl}}_2$ factors. Extending two homogeneous polynomial representations of SU2 therefore gives every finite-dimensional irreducible complex group representation of the Lorentz spinor double cover. The kernel sign permits descent precisely when $j_L+j_R$ is an integer. These field group representations are generally nonunitary for a positive-definite inner product; infinite-dimensional unitary Lorentz group representations belong to a different construction.