Image of linear K-derivations and linear ΚƐ-derivations of K[x1,x2,x3,x4]

Let K be a field of characteristic zero. We show that the image of a linear K-derivation of rank less than or equal to two over the polynomial algebra K[x1,x2,x3,x4] is a Mathieu-Zhao subspace. We have further shown that, under certain conditions, the image of KE-derivations of K[x1,x2,x3,x4] is a Mathieu-Zhao subspace.