Biquadratic morphism for elliptic sums and differences ($\Phi :{\mathbb{P}}^1\times {\mathbb{P}}^1\to {\mathbb{P}}^2$)

For a nonsingular Short Weierstrass form $y^2=x^3+ax+b$ over a number field, the unordered pair of coordinates $x(P+Q),x(P-Q)$ depends only on $x(P),x(Q)$. As a binary quadratic's roots, that pair is represented in ${\mathbb{P}}^2$ by the coefficientswith polynomial $C{T}^2-BT+A$. Homogenizing gives three forms of bidegree $(2,2)$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. They have no common zero. Off the diagonal $C\ne 0$; on the finite diagonal $x_1=x_2=t$, $B=4f(t)$ and $A=f^{\prime }(t{)}^2-8tf(t)$ for $f(t)=t^3+at+b$, so simultaneous vanishing would make a multiple root of $f$. At $(\infty ,\infty )$, the homogeneous $A$ is nonzero. Therefore they define a global morphism, also when a sum or difference is the identity.