Casimir operators for Poincare algebra I have seen at various places the comment that the operator $P_\mu P^\mu$ is a Casimir operator of Lorentz algebra and thus it satisfies a on-shell condition like $P_\mu P^\mu=m^2$. Given the Poincare algebra \begin{aligned}
i\left[M^{\mu \nu}, M^{\rho \sigma}\right] &=g^{\mu \sigma} M^{\nu \rho}+g^{\nu \rho} M^{\mu \sigma}-g^{\mu \rho} M^{\nu \sigma}-g^{\nu \sigma} M^{\mu \rho} \\
i\left[P^{\mu}, M^{\rho \sigma}\right] &=g^{\mu \rho} P^{\sigma}-g^{\mu \sigma} P^{\rho} \\
\left[P^{\mu}, P^{\nu}\right] &=0.
\end{aligned}
How does one derive its Casimir operators, especially the one $P_\mu P^\mu$? Can someone show the crucial steps? Also Does the method works for any other similar algebra? Moreover, if an operator, say, $A$ commutes with the generators $M^{\mu\nu}$, i.e., $[A,M^{\mu\nu}]=0$, can it be said that $A$ is a Casimir operator?
 A: Casimir operators commute with all generators. That's what you need to check. $P^\mu P_\mu$ does commute with $M$ and $P$. A fast way to say it is that


*

*$[P^2,P_\mu] \propto [P_\nu,P_\mu] =0$

*$P^2$ is a scalar and therefore it's annihilated by the rotation generators


But if you don't believe 2. you can just check
$$
\begin{aligned}
i[M_{\mu\nu},P^2] &= 2\,(g_{\rho\mu}P_\nu-g_{\rho\nu}P_\mu)P^\rho\\
&=2 P_\mu P_\nu - 2 P_\nu P_\mu \\&= 0\,.
\end{aligned}
$$
For other Casimirs such as
$$
W^\mu W_\mu\,,\qquad W_\mu := \tfrac12 \varepsilon_{\mu\nu\rho\lambda} M^{\nu\rho}P^\lambda\,,
$$
you can do the same. This is just a bit harder. The argument 2. still works because it's a scalar. Then by explicit computation
$$
[P_\mu,W_\nu] = 0\,,
$$
so also its square commutes with $P$.
A: As far as I know, there are no ways of deriving Casimirs. In the case of the Poincare algebra one guesses and checks that $P_{\mu}P^{\mu}$ and $W_{\mu}W^{\mu}$ are Casimirs, based on physical principles.
There are some ways of deriving the quadratic Casimir of a (simple) Lie algebra. If you calculate its Cartan metric then the quadratic Casimir is $g_{\mu \nu}x^{\mu}x^{\nu}$. 
You might check the answers to this question: Explicit Quadratic Casimir for $sp(2N)$
