A_{2m+1}$ can be expressed uniquely in the form $$\label{lincomb} \omega_{a,b} =\sum_{i=0}^m \lambda_i \partial^{2i}j^{2m-2i},\ \ \ \ \ \  \omega_{c,d} =\sum_{i=0}^m \mu_i \partial^{2i+1}j^{2m-2i}$$ for constants $\lambda_i,\mu_i$, $i=0,\dots,m$. Hence $\{j^{2m}|\ m\geq 0\}$ is an alternative strong generating set for $\cH(n)^{O(n)}$, and it will be convenient to pass back and forth between the sets $\{j^{2m}|\ m\geq 0\}$ and $\{\omega_{a,b}|\ 0\leq a\leq b\}$.

\[ordfingen\] $\cH(n)^{O(n)}$ is generated as a vertex algebra by $j^0$ and $j^2$.

Let $\cJ$ denote the vertex subalgebra of $\cH(n)^{O(n)}$ generated by $j^0$ and $j^2$. We need to show that $j^{2m}\in\cJ$ for all $m\geq 2$. Specializing (\[opeformula\]) shows that $$\label{specialope} j^2\circ_1 j^{2k} = 4 \omega_{2,2k} + (4+4k) j^{2k+2}.$$ Moreover, it is easy to check that $\omega_{2,2k} \equiv - j^{2k+2}$ modulo second derivatives. It follows that $j^2\circ_1j^{2k} \equiv (3+4k)j^{2k+2}$ modulo a linear combination of elements of the form $\partial^{2i} j^{2k+2-2i}$ for $1\leq i \leq k+1$. The claim then follows by induction on $k$.

Consider the category of all vertex algebras with generators $\{J^{2m}|~m\geq 0\}$, which satisfy the same OPE relations as the generators $\{j^{2m}|~m\geq 0\}$ of $\cH(n)^{O(n)}$. Since the vector space with basis $\{1\}\cup \{\partial^lj^{2m}|~l,m\geq 0\}$ is closed under all nonnegative circle products, it forms a Lie conformal algebra. By Theorem 7.12 