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?