Is there a nice description for the ring $\mathbb{C}[\mathfrak{g}]^G$, where $G$ is semisimple Some context for my question. Given a vector space $V$, $\mathbb{C}[V]$ is defined to be the ring of polynomials on the vector space $V$. In other words, we can make the identification $\mathbb{C}[V] = Sym V^*$, where $V^*$ is the dual vector space.
I wish to know whether there is a nice description for $\mathbb{C}[\mathfrak{g}]^G$.
An explicit description of the invariants in the case of $G = A_2$ would be enlightening and helpful too.
 A: Spivak, Comprehensive Introduction to Differential Geometry, volume 5, chapter 13, explains how to calculate these using classical invariant theory for all of the classical groups and gives the computations explicitly for $SO(n)$: the Pfaffian (if $n$ is even) and the even degree elementary symmetric functions of the eigenvalues. For $A_2$, the invariant polynomials are the  determinant and the trace of the square of each matrix, as the trace is zero. More generally for $A_n$ you get the elementary symmetric functions of the eigenvalues, except for the trace (which is zero), as in the theory of Chern classes.
I think you find all of the invariants written out pretty explicitly in M. L. Mehta, Basic sets of invariant polynomials for finite reflection groups, Comm. in Algebra, 16 (5), 1988, p. 1083-1098. The idea is that any invariant polynomial must restrict to any Cartan subalgebra to become invariant under the Weyl group, and conversely any Weyl group invariant polynomial extends to an invariant polynomial on the Lie algebra. 
