\bigoplus_{k\geq 0} V_k)^{O(n)}$ sending $Q_{a,b}\mapsto q_{a,b}$ is generated by the $(n+1)\times (n+1)$ determinants $$\label{weylrel} d_{I,J} = \det \left[\begin{matrix} Q_{i_0,j_0} & \cdots & Q_{i_0,j_n} \cr  \vdots  & & \vdots  \cr  Q_{i_n,j_0}  & \cdots & Q_{i_n,j_n} \end{matrix} \right].$$ In this notation, $I=(i_0,\dots, i_{n})$ and $J = (j_0,\dots, j_{n})$ are lists of integers satisfying $$\label{ijineq} 0\leq i_0<\cdots <i_n,\ \ \ \ \ \  0\leq j_0<\cdots <j_n.$$ Since $Q_{a,b} = Q_{b,a}$, it is clear that $d_{I,J} = d_{J,I}$.

Under the projection $$\phi_2: (\cH(n)^{O(n)})_{(2)}\ra (\cH(n)^{O(n)})_{(2)}/(\cH(n)^{O(n)})_{(1)}\subset gr(\cH(n)^{O(n)}) \cong (Sym \bigoplus_{k\geq 0} V_k)^{O(n)},$$ the generators $q_{a,b}$ of $(Sym \bigoplus_{k\geq 0} V_k)^{O(n)}$ correspond to vertex operators $\omega_{a,b}$ given by $$\label{omegagen} \omega_{a,b} = \sum_{i=1}^n :\partial^a \alpha^i \partial^b \alpha^i:,\ \ \ \ \ \ 0\leq a\leq b.$$ By Lemma \[recon\], the set $\{\omega_{a,b}|~0\leq a \leq b\}$ is a strong generating set for $\cH(n)^{O(n)}$. Note that $\omega_{0,0} = 2L$, where $L$ is the Virasoro element (\[virasoro\]). The subspace $(\cH(n)^{O(n)})_{(2)}$ of degree at most 2 has a basis $\{1\} \cup \{\omega_{a,b}\}$, and for all $n\geq 0$, the operators $\omega_{a,b}\circ_n$ preserve this vector space. It follows that every term in the OPE formula for $\omega_{a,b}(z) \omega_{c,d}(w)$ is a linear combination of these generators, so they form a Lie conformal algebra. We calculate that 