recall that $H$ is the kernel of the norm map from $L^\times/(L^\times)^2$ down to ${\mathbb Q}^\times/({\mathbb Q}^\times)^2$. Thus, ${{\rm Sel}}^{(2)}({\mathbb Q},J)$ is contained in the larger group $$\widehat{H}= \{ \xi \in L^\times/(L^\times)^2 : {{\rm val}}(\xi) \in G,\ \text{res}_\infty(\xi) \in J_\infty \}.$$ We emphasize here that the definition of $\widetilde{H}$ in [@stoll] does not impose a condition at $\infty$, but the definition of $\widehat{H}$ does to improve the bounds accuracy (thus $\widehat{H}\subseteq \widetilde{H}$). In an attempt to simplify notation, let $L_{J_\infty}$ be the subspace of $L^\times/(L^\times)^2$ with a condition added at infinite primes by $L_{J_\infty} = L^\times/(L^\times)^2 \cap {{\rm res}}_\infty^{-1}(J_\infty)$. Thus, $\widehat{H}=\{ \xi \in L_{J_\infty} : \text{val}(\xi) \in G\},$ and ${{\rm Sel}}^{(2)}({\mathbb Q},J)\subseteq \widehat{H}$. Note that $\widehat{H}$ is the largest such subgroup of $L_{J_\infty}$ such that $${{\rm val}}(\widehat{H})\cong  G\cap \text{val}\left(L_{J_\infty}\right).$$ Let us define subspaces $V$ and $W$ of $L^\times/(L^\times)^2$ as follows:

-   Let $\{\xi_i\}_{i=1}^r$ be generators of $G\cap \text{val}\left(L_{J_\infty}\right)$ with $\dim(G\cap \text{val}\left(L_{J_\infty}\right))=r$, and for each $1\leq i\leq r$ pick one $\mu_i\in L_{J_\infty}$ such that ${{\rm val}}(\mu_i)=\xi_i$. Let $W$ be the subspace generated by $\{\mu_i\}_{i=1}^r$. Note that $W\subseteq L_{J_\infty} \subseteq L^\times/(L^\times)^2$. In particular, ${{\rm res}}_\infty(w)\in J_\infty$ for all $w\in W$. Moreover, $W$ and ${{\rm val}}(W)$ are isomorphic by construction, so $$W\cong {{\rm 