works over an arbitrary field $K$ of characteristic not $2$ so long as every genus-$2$ curve over $K$ can be written in the form $y^2 = \text{(sextic)}$. This is the case for every field with more than $5$ elements.

The endomorphism of $D Y^2 = G_1 G_2 G_3$ whose existence is claimed by Theorem \[T-Bending\] is obtained by noting that the obvious Richelot dual of the curve is isomorphic over $K$ to the curve itself. Thus the degree-$4$ isogeny from the Jacobian of the curve to the Jacobian of its dual can be viewed as an endomorphism of the curve’s Jacobian, and this endomorphism has the properties claimed in the theorem.

We will want to consider curves over $K$ whose Jacobians have real multiplication by $\sqrt{2}$ that is not necessarily defined over $K$. For this reason, we will require the following variant of Bending’s construction:

Suppose $r$, $s$, and $t$ are elements of a field $K$ of characteristic not $2$, with $s\neq 0$, $s\neq 1$, and $t\neq 1$. Let $$\begin{aligned}
c_2 &= r + 4t\\
c_1 &= 4 t (r + s^3 - s^2 t - 2 s^2 + 5 s + t)\\
c_0 &= 4 t (s - 1) 