I have been thinking about this for some time but have had no luck. I have found some sources that say higher Casimir elements can be obtained by generalizing the second order Casimir, which is $\sum_{\alpha,\beta} \kappa ^{\alpha \beta} X_{\alpha} X_{\beta}$, where $\kappa ^{\alpha \beta}$ is the inverse of the Killing form, and writing $C_3 = \sum g^{\alpha_1 \alpha_2 \alpha_3} X_{\alpha_1} X_{\alpha_2} X_{\alpha_3}$, where $g^{\alpha_1 \alpha_2 \alpha_3} = Tr(adX^{\alpha_1} adX^{\alpha_2} adX^{\alpha_3})$ and $X^{\alpha} = \kappa^{\alpha \beta}X_{\beta}$. This definition does not give an element in the center of the universal enveloping algebra.
Is there any text out there where an explicit description of higher Casimir operators is given?
Best Answer
To answer the question in the title, it's $$(h_2-h_1)(h_2+2h_1+3)(2h_2+h_1+3) - 9f_1(h_1+2h_2+3)e_1 + 9f_2(2h_1+h_2+3)e_2 + 9f_{12}(h_2-h_1)e_{12} -27f_1f_2e_{12} -27f_{12}e_1e_2$$
where $e_{12}=[e_1,e_2]$ and $f_{12}=[f_2,f_1]$. You can find a version of this in a paper by Catoiu called Prime ideals of the universal enveloping algebra $U(\mathfrak{sl}_3)$. I don't know of any source that constructs the higher Casimirs explicitly, but I'm sure it's been discussed here before.