unique biquadratic subextension $K/F$ corresponding to the unique order 2 subgroup $\langle\sigma^2\rangle$ of $G$, so that ${\text{Gal}}(K/F)=G/\langle\sigma^2\rangle$. Let $s,t\in\{\sigma,\tau,\sigma\tau\}$ with $ s \neq t $, and let $\alpha,\beta$ be elements of $K$ such that $ \alpha^{2}, \beta^{2} \in F $, $s(\alpha)=\alpha, t(\alpha)=-\alpha, s(\beta)=-\beta$ and $t(\beta)=\beta$; note that $K=F(\alpha,\beta)$. We also fix an algebraic closure $F^{\text{alg}}$ of $ F $, and let $ \Omega = {\text{Gal}}(F^{\text{alg}}/F) $.

If $ N $ is abelian then $H= L[N]^{G} $ is a commutative separable $ F $-algebra, and hence, by [@WATERHOUSE §6.3], corresponds to a finite $ \Omega $-set. Specifically, $ L[N]^{G} $ corresponds to the $ \Omega $-set $ \widehat{N}  = \mbox{Hom}(N,F^{\text{alg}}) $, where $ \Omega $ acts on $ N $ by factoring through $ G $, and on $ \widehat{N} $ by $ \left(^{\omega}\chi \right)[\eta] = \omega(\chi(^{\omega^{-1}} \eta)) $ for all $ \eta \in N $ (in fact, the action of $ \Omega $ on $ \widehat{N} $ factors through $ {\text{Gal}}{(L^{\prime}/K)} $ for some cyclotomic extension $ L^{\prime} $ of $ L $). To make this correspondence explicit, let $\chi_1, \dots, \chi_s\in \widehat{N}$ be a set of representatives for the $\Omega $ orbits of $\widehat{N}$, and for each $i\in\{1,\ldots, s\}$ 