Is the Casimir element of $U(sl_2)$ equal to $ef+fe+h^2/2$ or $(h+1)^2/4+fe$? Is $ef+fe+h^2/2$ equal to $(h+1)^2/4+fe$? How to compute the Casimir element? I think that $ef+fe+h^2/2 = 2fe+(h+1)^2/2-1/2$. Thank you.
Casimir element of a universal enveloping algebra
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lie-algebras