= SU(3) Casimir in Chevalley generators
{c}
{title2=$C=3C_2$}
For $e_3=[e_1,e_2]$, $f_3=-[f_1,f_2]$, the operator $C=h_1^2+h_2^2+h_1h_2+3(h_1+h_2)+3\sum_{i=1}^3f_ie_i$ commutes with all generators. In its commutator with $e_1$, the Cartan polynomial contributes $3h_1e_1$, while the last sum contributes $-3h_1e_1$; the two mixed terms involving $f_2e_3$ cancel. On highest weight $(p,q)$ its eigenvalue is $p^2+q^2+pq+3p+3q$. This normalization is three times the conventional <SU(3) quadratic Casimir eigenvalue>.
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