I was asking this on stack exchange but I didn't get the answer. I am reading the proof in Borel's book, "Linear Algebraic Groups" of the fact if $G$ is connected affine group of dimension one, then it is either $\mathbb{G}_a$ or $\mathbb{G}_m$. In the proof it is written the translation extends uniquely to $\overline{G}$. Is this extension only as action of abstract groups or also as algebraic groups? What is the reason?

I was asking this on stack exchange but I didn't get the answer. Borel's book Linear Algebraic Groups contains the following result

10.9 Theorem. Let $G$ be a connected affine group of dimension one. Then $G$ is isomorphic to either $\mathbb{G}_a$ or $\mathbb{G}_1$.

Proof: $G$ is a dense open set in a unique complete smooth curve $\overline{G}$ (see AG.18.5(d)). It follows from (AG.18.5(f)) that the action of $G$ on itself by translation extends uniquely to an action of $G$ on $\overline{G}$...

Is the above extension of the action of $G$ as an action of abstract groups or also as algebraic groups. Why?