The Poincare group: rank and casimirs. It is often stated without proof in the particle physics literature that the Poincare group has rank $2$ and that as a result, the corresponding Lie algebra has exactly $2$ casimirs.
How can one show (preferably without the use of two many high-powered sledge hammer theorems) that the Poincare group has rank $2$?
Moreover, if one knows that the rank is $2$, how does it follow that the Poincare algebra has precisely two casimirs?  I know how to construct the two casimirs, but it's not clear to me how the dimension of a Cartan subalgebra restricts the number of casimirs.
 A: There are several notions of rank in the literature: Ordinary rank, reductive rank and semisimple rank. The first indeed equals 2 for Poincare group. The other two are equal to 1. 
Definition: Rank of a Lie algebra is the dimension of its maximal Cartan subalgebra. A subalgebra is Cartan it it is nilpotent and is self-normalizing. 
To see that 2 is a lower bound for the Poncare group consider a light-like line through the origin and its stabilizer in SO(1,3), which consists of elliptic rotations about this line. The subgroup of Poincare group generated by these two is abelian 2-dimensional. One then checks that it is self-normalizing. Checking that 2 is maximal possible dimensions for Cartans is a bit ugly case-by-case analysis. Let me know if you want to see it. 
In case of a semisimple Lie group (and Poincare group is not semisimple!) rank equals the dimensional of the space of Casimir elements (Racah theorem). I am not sure what happens in the non-semisimple case you care about, but a generalization of Racah theorem was proven for such groups in Theorem 15.1 in  
"Classical Groups for Physicists", by Brian G. Wybourne, 1974. 
I looked at his formula but it did not quite make sense to me, maybe you can figure it out (it might be the rank, but in some very odd form). 
