# Rank of the Poincare group

There are two Casimirs of the Poincare group:

$$C_1 = P^\mu P_\mu, \quad C_2 = W^\mu W_\mu$$

with the Pauli-Lubanski vector $W_\mu$. This implies the Poincare group has rank 2.

Is there a way to show that there really are no other Casimir operators other than trying to build all possible combinations of generators and seeing them fail?

To state it differently: Can one determine the rank of the Poincare group without explicit construction of the Casimir operators?

There is a way to "understand" why the number of Casimirs of the Poincaré group is 2. The Poincaré group is a Wigner-İnönü contraction of the de-Sitter group $SO(4,1)$ which is semisimple and of rank 2.
The Casimirs of the Poincaré group can be obtained from the Casimirs of $SO(4,1)$ explicitly in the contraction process. This is not a full proof because group contractions are singular limits, but at least it is a way to understand the case of the Poincaré group.