On the maximality of the triangular subgroup

We prove that the subgroup of triangular automorphisms of the complex affine $n$-space is maximal among all solvable subgroups of $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^n)$ for every $n$. In particular, it is a Borel subgroup of $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^n)$, when the latter is viewed as an ind-group. In dimension two, we prove that the triangular subgroup is a maximal closed subgroup. Nevertheless, it is not maximal among all subgroups of $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^2)$. Given an automorphism $f$ of $\mathbb{A}_{\mathbb{C}}^2$, we study the question whether the group generated by $f$ and the triangular subgroup is equal to the whole group $\mathrm{Aut}(\mathbb{A}_{\mathbb{C}}^2)$.