Quadratic homogeneous Keller maps of rank two

Let H be a quadratic homogeneous polynomial map of dimension n over an infinite field in which 2 is invertible such that its Jacobian JH is nilpotent. Meisters and Olech have shown that JH is strongly nilpotent if n≤4. They also proved that it is not true when n=5. We show that if rankJH≤2 and n arbitrary, then JH is strongly nilpotent. We also give examples to show that this is no longer true for any rank and dimension as long as the rank is greater than 2.