I have just started to read Poisson Geometry and was working my way through some problems. This may be basic for many, but I am just trying to understand the way to work through it. The problem says:
Consider the expressions in $\mathbb{R}^3$ for the Poisson bracket: $\{x_1,x_2\}=x_3$, $\{x_2,x_3\}=x_1$, $\{x_3,x_1\}=x_2$, $\{x_i,x_j\}=-\{x_j,x_i\}$.
(a) Verify this is really a Poisson bracket.
(b) Can it be symplectic? (i.e. non-degenerate)
(c) Find a nontrivial Casimir function in case it is not symplectic.
Parts (a) and (b) are straightforward, and it is not symplectic since we are in $\mathbb{R}^3$. I am a bit stuck as to how to compute the Casimir function in (c). What should be my line of thought here?
Thanks in advance.