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We compute by hand all quadratic homogeneous polynomial maps $H$ and allKeller maps of the form $x + H$, for which ${\rm rk} J H = 3$, over a field ofarbitrary 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 arbitrarycharacteristic: for Keller maps $x + H$ for which ${\rm rk} J H \le 4$, therows of $J H$ are dependent over the base field.

Quadratic homogeneous polynomial maps $H$ and Keller maps $x+H$ with $3 \le {\rm rk} J H \le 4$