Reductive subgroups of ${\mathrm Aut}{\mathbf A}^3$ and ${\mathrm Aut}{\mathbf A}^4$

We consider connected algebraic subgroups $G$ of the automorphism group of $n$-dimensional affine space and dwell on the linearization problem, i.e., that of conjugacy of $G$ with a subgroup of ${\rm GL}_n$. In particular, we prove that this conjugacy holds for $n=3$ and, if $G$ is not a one- or two-dimensional torus, for $n=4$.