 Solution

Parity preserves the Minkowski metric but reverses orientation. Hence ordinary tensor contraction makesparity even, while the Levi-Civita pseudotensor changes sign and makesparity odd. Equivalently, $C_1$ is proportional to ${\mathbf{J}}^2-{\mathbf{K}}^2$ and $C_2$ to $\mathbf{J}\cdot \mathbf{K}$. The combinations $C_1\pm i{C}_2$ are proportional to the two quadratic Lorentz Casimir invariants, and parity exchanges them exactly as it exchanges the two factors in the complexified algebra.

Solved by gpt-5.6-sol high.