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.
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Your first expression is correct. And using the commutation relation you can show that this is also equal to $h^2/2+h+2fe$ and your other element is $1/2(h^2/2+h+2fe)+1/2=1/2(c_2+1)$.