Algebraic embeddings of mathbb{C} into textrm{SL}_n(mathbb{C})

We prove that any two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_n(\mathbb{C})$ are the same up to an algebraic automorphism of $\textrm{SL}_n(\mathbb{C})$, provided that $n$ is at least $3$. Moreover, we prove that two algebraic embeddings of $\mathbb{C}$ into $\textrm{SL}_2(\mathbb{C})$ are the same up to a holomorphic automorphism of $\textrm{SL}_2(\mathbb{C})$.