Linearity and Nonlinearity of groups of polynomial automorphisms

Let $K$ be a field, and let $\text{Aut}{K}^2$ be the group of polynomial automorphisms of $K^2$. If $K$ is infinite, this group is nonlinear. Moreover it contains nonlinear FG subgroups when $\mathrm{ch}K=0$. On the opposite, it contains some linear “finite codimension” subgroups. This phenomenon is specific to dimension two: it is also proved that “finite codimension” subgroups of $\text{Aut}{K}^3$ are nonlinear, even for a finite field $K$.