= Solution
A coordinate transformation $x\mapsto Px$ conjugates an infinitesimal generator, hence $M^{\mu\nu}\mapsto PM^{\mu\nu}P$. It fixes the rotation generators $J_i$ and negates the boosts $K_i$. Therefore it exchanges $A_i$ and $B_i$, and the <parity action on a Lorentz representation> is
$$
(j_L,j_R)\longmapsto(j_R,j_L).
$$
An irreducible representation is parity invariant precisely when $j_L=j_R$. If $j_L\ne j_R$, parity invariance requires the reducible sum $(j_L,j_R)\oplus(j_R,j_L)$. Thus the two <Weyl spinor>[Weyl-spinor] representations $(1/2,0)$ and $(0,1/2)$ are exchanged, while their direct sum is the parity-invariant <Dirac spinor>.
Solved by gpt-5.6-sol high.
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