val}}(W) = G\cap \text{val}\left(L_{J_\infty}\right) \subseteq G\cap \text{val}\left(L^\times/(L^\times)^2\right)=\ker(G\to {{\rm Cl}}(L)/2{{\rm Cl}}(L)).$$ Thus, $r=\dim(W)=\dim(G\cap \text{val}\left(L_{J_\infty}\right))\leq  \dim \ker(G\to {{\rm Cl}}(L)/2{{\rm Cl}}(L)).$

-   Next, let us write $V=\ker({{\rm val}}\colon L_{J_\infty} \to I(L)/I(L)^2)$. It follows that $\widehat{H} = V \oplus W$ (note that ${{\rm val}}(V)$ is trivial, while ${{\rm val}}(w)$ is non-trivial for every $w\neq 0$ in $W$).

Let $V=\ker({{\rm val}}\colon L_{J_\infty} \to I(L)/I(L)^2)$, let $U=\left({\mathcal{O}}_L^\times/({\mathcal{O}}_L^\times)^2\right)\cap L_{J_\infty}$, and let ${{\rm Cl}}(L_{J_\infty})$ be the class group defined as follows $ {{\rm Cl}}(L_{J_\infty})=I(L)/P(L_{J_\infty})$, where $I(L)$ is the group of fractional ideals of $L$, and $P(L_{J_\infty})$ is the group of principal fractional ideals $\mathfrak{A}=(\alpha)$ with a generator such that ${{\rm res}}_\infty(\alpha)\in J_\infty$. Then, there is an exact sequence: $$0 \mapsto U  \to V \to {{\rm Cl}}(L_{J_\infty})[2] \to 0,$$ where $${{\rm Cl}}(L_{J_\infty})[2] = \{[\mathfrak{A}] \in {{\rm Cl}}(L_{J_\infty})[2] : \mathfrak{A}^2 = (\alpha) \text{ for some } \alpha \in L_{J_\infty}\}.$$

Consider $$\xymatrix{
& L^\times \cap {{\rm res}}_\infty^{-1}(J_\infty) \ar[rr]^2\ar[d]& & L^\times \cap {{\rm res}}_\infty^{-1}(J_\infty)\ar[rr]\ar[d] & & L_{J_\infty}\ar[rr]\ar[d]^{{\rm val}}& &0  \\
0 \ar[r] & I(L)\ar[rr]^2 & & I(L)\ar[rr] & &I(L)/I(L)^2\ar[rr]  & &0
}$$ and apply the snake lemma.

Let $P(L)$ be the subgroup of all principal fractional ideals, let $P^+(L)$ be the subgroup of principal ideals generated by a totally positive element, and 