= Finite-dimensional complex Lorentz representations
{title2=$D^{(j_L,j_R)}(S)=\operatorname{Sym}^{2j_L}S\otimes\operatorname{Sym}^{2j_R}\overline S$}
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.
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