  We classify Keller maps $x + H$ in dimension $n$ over fields with $\tfrac16$,
for which $H$ is homogeneous, and
  (1) deg $H = 3$ and rk $JH \le 2$;
  (2) deg $H = 3$ and $n \le 4$;
  (3) deg $H = 4$ and $n \le 3$;
  (4) deg $H = 4 = n$ and $H_1, H_2, H_3, H_4$ are linearly dependent over $K$.
  In our proof of these classifications, we formulate (and prove) several
results which are more general than needed for these classifications. One of
these results is the classification of all homogeneous polynomial maps $H$ as
in (1) over fields with $\tfrac16$.
