Quadratic linear Keller maps of nilpotency index three

Let f=x+H:Cn→Cn be a homogeneous polynomial map of degree d⩾2 and FA=y+(Ay)(d):CN→CN a power linear map such that f and FA are a generalized Gorni-Zampieri pair. We discuss the relation between the nilpotency indices of JH and J(Ay)(d) and we show that f is linearly triangularizable if and only if FA is linearly triangularizable. As a consequence, we show that a quadratic linear Keller map FA=y+(Ay)(2) with nilpotency index three, i.e., (J(Ay)(2))3=0, is linearly triangularizable.