of [@BK], this category possesses a universal object $\cV_n$, which is [*freely*]{} generated by $\{J^{2m}|\ m\geq 0\}$. In other words, there are no nontrivial normally ordered polynomial relations among the generators and their derivatives in $\cV_n$. Then $\cH(n)^{O(n)}$ is a quotient of $\cV_n$ by an ideal $\cI_n$, and since $\cH(n)^{O(n)}$ is a simple vertex algebra, $\cI_n$ is a maximal ideal. Let $\pi_n: \cV_n\ra \cH(n)^{O(n)}$ denote the quotient map, which sends $J^{2m}\mapsto j^{2m}$. Using the formula (\[lincomb\]), which holds in $\cH(n)^{O(n)}$ for all $n$, we can define an alternative strong generating set $\{\Omega_{a,b}| ~0\leq a\leq b\}$ for $\cV_n$ by the same formula: for $a+b = 2m$ and $c+d = 2m+1$, $$\Omega_{a,b} =\sum_{i=0}^m \lambda_i \partial^{2i}J^{2m-2i},\ \ \ \ \ \  \Omega_{c,d} =\sum_{i=0}^m \mu_i \partial^{2i+1}J^{2m-2i}.$$ Clearly $\pi_n(\Omega_{a,b}) = \omega_{a,b}$. We will use the same notation $A_m$ to denote the linear span of $\{\Omega_{a,b}|~ a+b = m\}$, when no confusion can arise. Finally, $\cV_{n}$ has a good increasing filtration in which $(\cV_n)_{(2k)}$ is spanned by iterated Wick products of the generators $J^{2m}$ and their derivatives, of length at most $k$, and $(\cV_n)_{(2k+1)} = (\cV_n)_{(2k)}$. Equipped with this filtration, $\cV_n$ lies in the category $\cR$, and $\pi_n$ is a morphism in $\cR$.

Since the vertex 