assume that $h$ is separable. We use the roots $\beta_1$, $\beta_2$, $\beta_3$ of $h$ to define three polynomials $g_1$, $g_2$, $g_3$ in the polynomial ring ${{K\llap{$\overline{\phantom{\rmK}}$}}}[x]$, we assume that the product $f = g_1 g_2 g_3$ is separable, and we define a curve $C$ by $y^2 = f$. Then we show that the Richelot dual of $C$ corresponding to the factorization $f = g_1 g_2 g_3$ is geometrically isomorphic to $C$ itself.

Let $L$ be the quotient of the polynomial ring $K[T]$ by the ideal generated by the polynomial $h$ and let $\beta$ be the image of $T$ in $L$. Since $h$ is separable, the algebra $L$ is a product of fields. Let $g\in L[x]$ be the polynomial $$g = x^2 - 2\beta x + (1-s)\beta^2 - 4 s (s-1)^2 t (s - t - 1).$$ Let $\Delta\in K^*$ be the discriminant of $h$ and let $\Delta'\in L^*$ be the discriminant of $g$.

\[T-Galois\] There are distinct Galois-stable maximal isotropic subgroups $G$ and $G'$ of $({\operatorname{Jac}}C)[2]$ with $G' = (1+\sqrt{2})(G)$ if and only if $\Delta\Delta'$ is a square in the algebra $L$.

Let the roots of $g_1$ (respectively, $g_2$, $g_3$) be $r_1$ and $r_2$ (respectively, $r_3$ and $r_4$, $r_5$ and 