  Let $\mathcal{G}$ be an ind-group and let $\mathcal{U} \subseteq \mathcal{G}$
be a unipotent ind-subgroup. We prove that an abstract group automorphism
$\theta \colon \mathcal{G} \to \mathcal{G}$ maps $\mathcal{U}$ isomorphically
onto a unipotent ind-subgroup of $\mathcal{G}$, provided that $\theta$ fixes a
closed torus $T \subseteq \mathcal{G}$, which normalizes $\mathcal{U}$ and the
action of $T$ on $\mathcal{U}$ by conjugation fixes only the neutral element.
As an application we generalize a result by Hanspeter Kraft and the author as
follows: If an abstract group automorphism of the affine Cremona group
$\mathcal{G}_3$ in dimension 3 fixes the subgroup of tame automorphisms
$T{\mathcal G}_3$, then it also fixes a whole family of non-tame automorphisms
(including the Nagata automorphism).
