SU(3) Casimir in Chevalley generators ($C=3C_2$)

For $e_3=[e_1,e_2]$, $f_3=-[f_1,f_2]$, the operator $C=h_1^2+h_2^2+h_1{h}_2+3(h_1+h_2)+3{\sum }_{i=1}^3{f}_i{e}_i$ commutes with all generators. In its commutator with $e_1$, the Cartan polynomial contributes $3h_1{e}_1$, while the last sum contributes $-3h_1{e}_1$; the two mixed terms involving $f_2{e}_3$ 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.