Chiral decomposition of the complex Lorentz algebra

Over the complex numbers, $A_i=(J_i+i{K}_i)/2$ and $B_i=(J_i-i{K}_i)/2$ generate commuting copies of ${\mathfrak{sl}}_2(\mathbb{C})$, givingFinite-dimensional irreducible representations are labelled $(j_L,j_R)$.

For Hermitian rotations $J_i$ and boosts $K_i$, the complex combinations $A_i=(J_i+i{K}_i)/2$ and $B_i=(J_i-i{K}_i)/2$ are adjoints of one another. Although their complex brackets split into two commuting ${\mathfrak{sl}}_2$ factors, they are not two independent Hermitian compact rotation algebras. The real Lorentz spin group is $SL(2,\mathbb{C})$; $SU(2)\times SU(2)$ is a different compact real form.

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.

Parity fixes $J_i$, negates $K_i$, and therefore exchanges the two chiral factors. It sends $(j_L,j_R)$ to $(j_R,j_L)$.

The quadratic contraction $M_{\mu \nu }{M}^{\mu \nu }$ is parity even, while $M_{\mu \nu }{\tilde{M}}^{\mu \nu }$ is parity odd. Their linear combinations give the quadratic Casimirs of the two chiral ${\mathfrak{sl}}_2$ factors.