  We compute by hand all quadratic homogeneous polynomial maps $H$ and all
Keller maps of the form $x + H$, for which ${\rm rk} J H = 3$, over a field of
arbitrary characteristic.
  Furthermore, we use computer support to compute Keller maps of the form $x +
H$ with ${\rm rk} J H = 4$, namely:
  $\bullet$ all such maps in dimension $5$ over fields with $\frac12$;
  $\bullet$ all such maps in dimension $6$ over fields without $\frac12$.
  We use these results to prove the following over fields of arbitrary
characteristic: for Keller maps $x + H$ for which ${\rm rk} J H \le 4$, the
rows of $J H$ are dependent over the base field.
