Consider some $a,b,c\in E$. Then $a\oplus b\oplus c=0_E$ in the group $E$ iff $a+b+c = 3\cdot 0_E$ in $\mathrm{Pic}^3(E)$ iff $a,b,c$ are colinear in the complete linear system $|\mathcal{O}_E(3\cdot 0_E)|\cong\mathbb{P}^2$. This is true over any field.
David Lehavi
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