Let $$E:Y^2Z=X^3+AXZ^2+BZ^3$$ be an elliptic curve with $A,B\in \mathbb{Z}$. I would like to understand the map \begin{align} G:E\times E &\to E\times E\\ (P,Q)&\mapsto (P+Q,P-Q). \end{align} With the Segre embedding $\sigma$, I identify $E\times E$ with a closed subset of $\mathbb{P}^8$. Concretely, $\sigma$ is the map $$\big([X_P,Y_P,Z_P],[X_Q,Y_Q,Z_Q]\big )\mapsto \begin{bmatrix}X_PX_Q & Y_PX_Q & Z_PX_Q\\ X_PY_Q & Y_PY_Q & Z_PY_Q\\ X_PZ_Q & Y_PZ_Q & Z_PZ_Q\end{bmatrix}.$$ Now, I believe it should be possible to calculate homogeneous polynomials in the "Segre coordinates" $$g_{11},\dots,g_{33}\in\mathbb{Z}[X_PX_Q,Y_PX_Q,\dots,Z_PZ_Q]$$ such that if $g=[g_{ij}]$ we have $$(P+Q,P-Q)=g\big([X_P,Y_P,Z_P],[X_Q,Y_Q,Z_Q]\big).\tag{$\star$}$$
My question is: what is the lowest possible degree of the $g_{ij}$?
Unless I have miscalculated, we have $$\begin{bmatrix}X_{P+Q}\\ Y_{P+Q}\\ Z_{P+Q}\end{bmatrix}= \begin{bmatrix} A(X_P^2Z_Q^2-Z_P^2X_Q^2)+3B(X_PZ_PZ_Q^2-Z_P^2X_QZ_Q)+Y_P^2X_QZ_Q-2X_PY_PY_QZ_Q\dots\\ \quad \dots+2Y_PZ_PX_QY_Q-X_PZ_PY_Q^2\\[0.5em] 3(X_P^2X_QY_Q-X_PY_PX_Q^2)+Y_P^2Y_QZ_Q-Y_PZ_PY_Q^2-A(X_PY_PZ_Q^2+2Y_PZ_PX_QZ_Q\dots\\ \quad \dots-2X_PZ_PY_QZ_Q-Z_P^2X_QY_Q)-3B(Y_PZ_PZ_Q^2-Z_P^2Y_QZ_Q)\\[0.5em] Y_P^2Z_Q^2-Z_P^2Y_Q^2 - A(X_PZ_PZ_Q^2-Z_P^2X_QZ_Q) - 3(X_P^2X_QZ_Q-X_PZ_PX_Q^2) \end{bmatrix},$$ and a similar equality holds for $P-Q$. It is not hard to see that these are quadratic in the Segre coordinates. The image $g(P,Q)$ should have coordinates equal to $$\begin{bmatrix}X_{P+Q}X_{P-Q} & Y_{P+Q}X_{P-Q} & Z_{P+Q}X_{P-Q}\\ X_{P+Q}Y_{P-Q} & Y_{P+Q}Y_{P-Q} & Z_{P+Q}Y_{P-Q}\\ X_{P+Q}Z_{P-Q} & Y_{P+Q}Z_{P-Q} & Z_{P+Q}Z_{P-Q} \end{bmatrix},$$ and so we can use the formulas for $P\pm Q$ to find a representative for this image. But now the degrees are $4$ (in the Segre coordinates). It is a theorem that $$H\big((P+Q,P-Q)\big)=O(H(P)^2H(Q)^2),$$ (here $H$ is the standard multiplicative height) which makes one hope that it is possible to find another representative for $(P+Q,P-Q)$ with coordinates that are quadratic in the Segre coordinates. Maybe this is a trivial question, I am new to these kinds of computations.