Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see
Lange and Ruppert - Complete systems of addition laws on abelian varieties,
Bosma - Complete systems of two addition laws for elliptic curves). It is known that the group law on $E$ can be represented by bi-homogenous polynomials of bidegree $(2, 2)$. More precisely, those articles show that for the divisor $D := 3(2V + 2H -\triangledown)$ the global section space $H^{0}(E^2, D)$ is non-zero, where $\triangledown$ is the anti-diagonal, $V := E \times \{\mathcal{O}\}$, and $H := \{\mathcal{O}\} \times E$. Moreover, $\lvert D\rvert$ is a base-point free linear system.

I am interested in the addition of three points $P := P_0 + P_1 + P_2$ on $E$. Of course, $P = (P_0 + P_1) + P_2$. However, expressing $P$ in this way gives rise to tri-homogenous polynomials of tridegree $(4,4,2)$. Alternatively, I guess that there are tri-homogenous polynomials of tridegree $(3,3,3)$. Since $3\cdot 3 = 9 < 10 = 2\cdot 4+2$, it seems that the latter polynomials can be evaluated more efficiently. This may have applications in elliptic cryptography.

Now let $D := 3(3A_0 + 3A_1 + 3A_2 - B)$, where $$ A_0 := \{\mathcal{O}\} \times E^2, \quad A_1 := E \times \{\mathcal{O}\} \times E, \quad A_2 := E^2 \times \{\mathcal{O}\}, \quad B := \big\{(P_0, P_1, -P_0-P_1)\big\} \quad \subset \quad E^3. $$ If I am not mistaken, to confirm my conjecture it is necessary to prove that the space $H^{0}(E^3, D)$ is non-zero. Help me please to do this. And what about base-point freeness of $\lvert D\rvert$?