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This is an answer to the question about the general structure of the centre of the enveloping algebra:

The centre of the enveloping algebra (in particular when $\mathfrak g$ is reductive) is one of the basic objects in the study of infinite-dimensional representations of Lie algebras and Lie groups. It was first described in general (for reductive $\mathfrak g$) by Harish-Chandra, as far as I know, who proved the following: let $\mathfrak g$ be a reductive Lie algebra (over $\mathbb C$, say) and let $\mathfrak h$ be a Cartan subalgebra. Let $W$ be the Weyl group, which acts on $\mathfrak h$ via the adjoint action.

Then there is an isomorphism $Z(\mathfrak g) \cong Sym(\mathfrak h)^W.$ (Here $Z(\mathfrak g)$ denotes the centre of the enveloping algebra $U(\mathfrak g)$, and $Sym(\mathfrak h)$ denotes the symmetric algebra of $\mathfrak h,$ which is the same as the enveloping algebra $U(\mathfrak h)$, since $\mathfrak h$ is abelian.)

E.g. if $\mathfrak g = {\mathfrak sl}_2$, then a Cartan subalgebra is one-dimensional, therefore $Sym(\mathfrak h)^W$ turns out to be a polynomial ring in one generator, where this generator can be taken to be the Casimir.

If $\mathfrak g$ is semi-simple of rank $l$, then the centre turns out to be a polynomial ring in $l$ generators, one of which is the Casimir, and the others of which can be taken to be the so-called higher Casimirs. (Their degrees are the so-called exponents of $\mathfrak g$, or perhaps the exponents shifted by one; I'm unsure about the standard normalization.)normalization. [Added: As Mike Skirvin confirms in a comment below, they are the exponents shifted by $1$.])

To learn more you can google Harish-Chandra isomorphism, or look in one of the many representation-theory texts that are out there. (I like Knapp's book Representation Theory of Semisimple Groups: An Overview Based on Examples, but there are lots of choices.)

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This is an answer to the question about the general structure of the centre of the enveloping algebra:

The centre of the enveloping algebra (in particular when $\mathfrak g$ is reductive) is one of the basic objects in the study of infinite-dimensional representations of Lie algebras and Lie groups. It was first described in general (for reductive $\mathfrak g$) by Harish-Chandra, as far as I know, who proved the following: let $\mathfrak g$ be a reductive Lie algebra (over $\mathbb C$, say) and let $\mathfrak h$ be a Cartan subalgebra. Let $W$ be the Weyl group, which acts on $\mathfrak h$ via the adjoint action.

Then there is an isomorphism $Z(\mathfrak g) \cong Sym(\mathfrak h)^W.$ (Here $Z(\mathfrak g)$ denotes the centre of the enveloping algebra $U(\mathfrak g)$, and $Sym(\mathfrak h)$ denotes the symmetric algebra of $\mathfrak h,$ which is the same as the enveloping algebra $U(\mathfrak h)$, since $\mathfrak h$ is abelian.)

E.g. if $\mathfrak g = {\mathfrak sl}_2$, then a Cartan subalgebra is one-dimensional, therefore $Sym(\mathfrak h)^W$ turns out to be a polynomial ring in one generator, where this generator can be taken to be the Casimir.

If $\mathfrak g$ is semi-simple of rank $l$, then the centre turns out to be a polynomial ring in $l$ generators, one of which is the Casimir, and the others of which can be taken to be the so-called higher Casimirs. (Their degrees are the so-called exponents of $\mathfrak g$, or perhaps the exponents shifted by one; I'm unsure about the standard normalization.)

To learn more you can google Harish-Chandra isomorphism, or look in one of the many representation-theory texts that are out there. (I like Knapp's book Representation Theory of Semisimple Groups: An Overview Based on Examples, but there are lots of choices.)