operators $J^{2m}$ satisfy the same OPE relations as the $j^{2m}$, $\cV_n$ is also generated as a vertex algebra by $J^0$ and $J^2$.

Recall the variables $Q_{a,b}$ and $q_{a,b}$ appearing in Theorem \[weylfft\]. Since $\cV_n$ is freely generated by $\{J^{2m}|\ m\geq 0\}$, and $\{\Omega_{a,b}|\ 0\leq a\leq b\}$ and $\{\partial^k J^{2m}|\ k,m\geq 0\}$ form bases for the same space, we may identify $gr(\cV_n)$ with $\mathbb{C}[Q_{a,b}]$, and we identify $gr(\cH(n)^{O(n)})$ with $\mathbb{C}[q_{a,b}]/I_n$. Under this identification, $gr(\pi_{n}): gr(\cV_n) \ra gr(\cH(n)^{O(n)})$ is just the quotient map sending $Q_{a,b}\mapsto q_{a,b}$. Clearly the projection $\pi_{n}: \cV_n\ra \cH(n)^{O(n)}$ maps each filtered piece $(\cV_n)_{(k)}$ onto $(\cH(n)^{O(n)})_{(k)}$, so the hypotheses of Lemma \[idealrecon\] are satisfied. Since $I_{n} = Ker (gr(\pi_{n}))$ is generated by the determinants $d_{I,J}$, we can apply Lemma \[idealrecon\] to find vertex operators $D_{I,J}\in (\cV_{n})_{(2n+2)}$ satisfying $\phi_{2n+2}(D_{I,J}) = d_{I,J}$, such that $\{D_{I,J}\}$ generates $\cI_{n}$. Since $\Omega_{a,b}$ has weight $a+b+2$, $$\label{wtod} wt(D_{I,J}) = |I| + |J| +2n+2,\ \ \ \ \ \ |I| =\sum_{a=0}^{n} i_a,\ \ \ \ \ \ |J| =\sum_{a=0}^{n} j_a.$$

In general, the vertex operators $a_i(z)$ furnished by Lemma \[idealrecon\] satisfying $\phi_{d_i}(a_i(z)) = a_i$ which generate $\cI$ are not unique. However, in our case, $D_{I,J}$ is uniquely determined by the conditions $$\label{uniquedij} \phi_{2n+2}(D_{I,J}) = d_{I,J},\ \ 