Semifield planes of order q4q4 with kernel Fq2Fq2 and center FqFq

A classification of semifield planes of order q4q4 with kernel Fq2Fq2 and center FqFq is given. For qq an odd prime, this proves the conjecture stated in [M. Cordero, R. Figueroa, On the semifield planes of order 54 and dimension 2 over the kernel, Note Mat. (in press)]. Also, we extend the classification of semifield planes lifted from Desarguesian planes of order q2q2, qq odd, obtained in [M. Cordero, R. Figueroa, On some new classes of semifield planes, Osaka J. Math. 30 (1993) 171–178], to the even characteristic case.