Let $H,X,Y$ be the standard basis of the Lie algebra $\mathfrak g = sl(2,\mathbb C)$. Up to a factor, the most common definition of the Casimir operator of a $\mathfrak g$-module is $c = h^2+2h+4yx$. (I write small letters for the corresponding operators on the $\mathfrak g$-module) Now, in several books (e.g. Varadarajan,Introduction to harmonic Analysis on Semisimple Lie groups) I also saw the definition $c = (h+1)^2 + 4 yx$ This divergence of course leads to other scalars with which these operators act on the representations of interest. So I wonder what the connection between the two definition is and what is it about?
I just realized that it is completely trivial that if an element $c$ is in the center of an algebra, then so is the element $c+1$ (or for that matter any $c+c'$ with $c'$ another central element. So my question is probably just why some authors prefer to use the second definition above. The first definition seems to be the canonical one if one uses the general definition of a Casimir element and the Cartan Killing form as nondegenerate bilinear form as in https://en.wikipedia.org/wiki/Casimir_element (if I compute correctly one actually gets $c = 1/8 h^2 + 1/4 h + 1/2 yx$ but everyone tends to multiply by $8$ here).