Algebraic embeddings of $\mathbb{C}$ into ${\text{SL}}_n(\mathbb{C})$

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