Symmetric polynomials in the variety generated by Grassmann algebras

Let $\mathcal{G}$ denote the variety generated by infinite dimensional Grassmann algebras; i.e., the collection of all unitary associative algebras satisfying the identity $[[z_1,z_2],z_3]=0$, where $[z_i,z_j]=z_i{z}_j-z_j{z}_i$. Consider the free algebra $F_3$ in $\mathcal{G}$ generated by $X_3=\{x_1,x_2,x_3\}$. The commutator ideal $F_3^{\mathrm{\prime }}$ of the algebra $F_3$ has a natural $K[X_3]$-module structure. We call an element $p\in F_3$ symmetric if $p(x_1,x_2,x_3)=p(x_{\xi 1},x_{\xi 2},x_{\xi 3})$ for each permutation $\xi\in S_3$\mathrm{.}Symmetricelementsformthesubalgebra$F_3^{S_3}$ of invariants of the symmetric group $S_3$ in $F_3$. We give a free generating set for the $K[X_3{]}^{S_3}$-module $(F_3^{\mathrm{\prime }}{)}^{S_3}$.