  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
$\textrm{SL}_2(\mathbb{C})$.
