Let $k$ be a field, and let $E \subset \mathbb{P}^2_k$ be a smooth projective plane cubic. Let $O \in E(k)$ be a $k$-rational flex point. For $P,Q \in E(k)$, let $P \circ Q \in E(k)$ denote the third intersection point of $E$ with the projective line through $P$ and $Q$, where the tangent line to $E$ at $P$ is used when $P = Q$, and all intersections are counted with their intersection multiplicities.

Define the map

\begin{align*} \iota: E(k) &\to E(k) \end{align*}

as follows: $\iota(O) = O$, and for $S \in E(k)$ with $S \ne O$, $\iota(S)$ is the third intersection point of $E$ with the projective line through $S$ and $O$, counted with multiplicity. Define a binary operation

\begin{align*} \oplus: E(k) \times E(k) \to E(k). \end{align*}

It is given by the rule

\begin{align*} \oplus(P,Q)=\iota(P \circ Q). \end{align*}

Then $O$ is the identity element for $\oplus$, and $\oplus$ is associative:

\begin{align*} (P \oplus Q) \oplus R = P \oplus (Q \oplus R) \end{align*}

for all $P,Q,R \in E(k)$.