special relativity - Heuristic derivation of $W^mu=frac{1}{2}epsilon^{munusigmarho}P_nu J_{sigmarho}$ using combination of physical and mathematical arguments

If a simple systematic way to derive or guess (either mathematically or by a combination of physical arguments and mathematics) that one of the Casimir operator of Poincare group is $W^2\equiv W_\mu W^\mu$ where $$W^\mu=\frac{1}{2}\epsilon^{\mu\nu\sigma\rho}P_\nu J_{\sigma\rho}\tag{1}.$$ In physics textbooks, (1) is given as a definition, and from which one can check that $W^2\equiv W_\mu W^\mu$ is indeed a Casimir. But I find this definition of $W^\mu$ to be quite non-trivial to guess. So I'm not looking for a rigorous derivation and if there are physical arguments to achieve this, it will do for me.