Degree-two and degree-three Harish-Chandra generators for ${\mathfrak{s}\mathfrak{l}}_3$

For diagonal $h=\mathrm{diag}(t_1,t_2,t_3)$, the root vectors satisfy $[h,E_{ij}]=(t_i-t_j)E_{ij}$. The corresponding root reflections from Root reflections and the Weyl group action interchange $t_i$ and $t_j$, so $W=S_3$ on the trace-zero Cartan. The symmetric polynomials in $t_1,t_2,t_3$ are generated freely by the elementary symmetric functions $e_1,e_2,e_3$. The trace-zero relation sets $e_1=0$. Every invariant class modulo this relation has an invariant representative, obtained by averaging any representative over the finite group $S_3$. If an invariant polynomial is a multiple of $e_1$ in the full polynomial ring, averaging its quotient makes it a multiple of $e_1$ already in the invariant ring. Thus the quotient invariant ring is exactly $\mathbb{C}[e_2,e_3]$. Newton's identities give $s_2=-2e_2$ and $s_3=3e_3$ there. Hence $s_2,s_3$ are algebraically independent generators.

Under Choice, The center of the enveloping algebra is polynomial on rank-many generators says the center of $U({\mathfrak{s}\mathfrak{l}}_3)$ is polynomial on two generators of definite PBW degrees. The filtered isomorphism Harish-Chandra isomorphism for the center and step 1.1 show that the preimages of $s_2$ and $s_3$ are algebraically independent generators of respective PBW degrees two and three.