$\gamma $. As in the above example, one solution is given by $\gamma = f.$ However, the inner bracket in this example [*can*]{} be degenerate, and thus have Casimir invariants of its own. Suppose there are $m$ such independent functions and denote them by $%
C^{(i)}.$ Then a solution of (\[INFNULL\]) is $\gamma =
k(C^{(1)},\ldots,C^{(m)},f),$ where $k$ is an arbitrary function. From this we see that

$$\label{GENINFNULL}
\Gamma = k(C^{(1)},\ldots,C^{(m)},f) \delta f$$

is a null covector of J. Not only that, it is an exact variation, making $%
\Gamma = 0$ easy to integrate. Indeed, we find as a Casimir invariant

$$\label{INFCAS}
{\cal C}[f] = \int _{D} K(C^{(1)},...,C^{(m)},f) {\rm d}^nz$$

where $k(f) =\partial K(f)/\partial f.$

Applying this result to the proposed spin gas bracket, we use the result from Section \[S-FINITE\], and obtain the Casimir invariant

$$\label{INFSO3CAS}
{\cal C}[f] = \int _{D} K((z^1)^2 + (z^2)^2 + (z^3)^2,f) {\rm d}^3z \, .$$

We could also envision a kinetic theory describing particles evolving on the Lie algebra su(3). Its Casimir invariant would be: $$\label{INFSU3CAS}
{\cal C}[f] = \int _{D} K(C_1, C_2,f) {\rm d}^8z \, ,$$ where $C_1, C_2$ are given respectively by equations (\[SU3-5\], \[SU3-9\]). In both (\[INFSO3CAS\]) and (\[INFSU3CAS\]), $K$ is 