let $P(L_{J_\infty})$ be as above. Since the trivial signature $(1,1,\ldots,1)\in J_\infty$, it implies that $P^+(L) \subseteq P(L_{J_\infty}) \subseteq P(L)$, and therefore there are surjections $${{\rm Cl}}(L) \twoheadleftarrow {{\rm Cl}}(L_{J_\infty}) \twoheadleftarrow {{\rm Cl}}^+(L),$$ where ${{\rm Cl}}^+(L)$ is the narrow class group of $L$.

Putting all this together (and writing $\dim$ for $\dim_{{\mathbb F}_2}$), we obtain a bound $$\begin{aligned}
\label{newbound}
\nonumber \dim{{\rm Sel}}^{(2)}({\mathbb Q},J)& \leq \dim(\widehat{H}) = \dim U + \dim {{\rm Cl}}(L_{J_\infty})[2] + \dim W \\ 
\nonumber & = \dim \left({\mathcal{O}}_L^\times/({\mathcal{O}}_L^\times)^2\cap {{\rm res}}_\infty^{-1}(J_\infty)\right) + \dim {{\rm Cl}}(L_{J_\infty})[2] + \dim G\cap \text{val}\left(L_{J_\infty}\right)\\
 & \leq \dim \left({\mathcal{O}}_L^\times/({\mathcal{O}}_L^\times)^2\cap {{\rm res}}_\infty^{-1}(J_\infty)\right)  + \dim {{\rm Cl}}^+(L)[2] + \dim G\cap \text{val}\left(L_{J_\infty}\right).\end{aligned}$$ We note here that $\dim \left({\mathcal{O}}_L^\times/({\mathcal{O}}_L^\times)^2\cap {{\rm res}}_\infty^{-1}(J_\infty)\right)\leq m_\infty -1$, where we have used the fact that $\dim({\mathcal{O}}_L^\times/({\mathcal{O}}_L^\times)^2)=m_\infty$, and the fact that ${{\rm res}}_\infty(-1)$ is not in $J_\infty$ (because $J_\infty=\delta_\infty(J({\mathbb R}))\subseteq H_\infty$, which is the kernel of the norm map, so $N(j)=1$ for $j\in J_\infty$, but the norm $N(-1)=-1$ because the degree of $L$ is odd). We will improve on the bound given by (\[newbound\]) above by making certain assumptions about $G$ and a more careful analysis of the dimension of the subgroup of totally positive units. Before we state our refinements, we review some of 