\nonumber \\
&& - 6\sqrt{3} (z^2)^2 z^8 -9 z^3 (z^4)^2 - 9 z^3 (z^5)^2 + 9 z^3 (z^6)^2 +
9 z^3 (z^7)^2  \label{SU3-9} \\
&& - 6 \sqrt{3} (z^3)^2 z^8 + 3 \sqrt{3} (z^4)^2 z^8 + 3 \sqrt{3} (z^5)^2
z^8 + 3 \sqrt{3} (z^6)^2 z^8  \nonumber \\
&& + 3 \sqrt{3} (z^7)^2 z^8 + 2 \sqrt{3} (z^8)^3 \, .  \nonumber\end{aligned}$$

We should note here that for Lie-Poisson brackets based on semi-simple Lie algebras (as in this example), a formula exists for the Casimir invariants. It follows immediately from results in [@gror] that if the basis elements of the algebra are given as matrices $X^{(i)}$ in any representation of the Lie algebra, then $C = Tr(X^{(i)} \cdots X^{(j)}) z^i
\cdots z^j$ is a Casimir invariant (here, $z^i$ is the coordinate dual to the basis vector $X^{(i)}$). Using the Gell-Mann matrices in this formula yields Casimir invariants proportional to (\[SU3-5\]) and (\[SU3-9\]).

As complicated as this last example might seem, the method used is certainly more economical than solving the system of coupled PDEs mentioned in Section \[S-FINITE\], which is the only systematic alternative known for the computation of Casimir invariants. And while for a small number of PDEs, solutions can often 