I am a Physicist, so let me apologize in advance for some possible imprecisions.

I'm working on a 10-dimensional Lie Algebra. Each element of the algebra represents a quantum mechanical operator, and I would like to find the "constants of motions" of my system. These, in turn, correspond to the Casimir operators of the algebra.

First of all: the number of Casimir operators corresponds to the rank of the Algebra, doesn't it? And the rank of the algebra is the maximum number of operators that commute among themselves, isn't it?

I've already written the matrix realization of my algebra and now I'm looking for an "algorithmic" way of findining the Casimir operators. I know my question sounds similar to this one: How to find Casimir operators?, yet I haven't understood it fully. Can you please give a detailed and practical guide on how to find them? If you wish, you can use the easy algebra $\mathbf{su}(2)$, pretending not to already know the final result.

index, say $\ell$; however, it is no longer true that the invariant operators are freely generated by $\ell$ (algebraically) independent "Casimir elements". $\endgroup$ – Victor Protsak Feb 22 '17 at 4:30everythingin $U(L)$, and not just with each other. $\endgroup$ – Victor Protsak Feb 22 '17 at 4:38