On the Hughes-Kleinfeld and Knuth's semifields two-dimensional over a weak nucleus

In 1960 Hughes and Kleinfeld \cite{HK1960} constructed a finite semifield which is two dimensional over a weak nucleus. In 1965 Knuth \cite{Knuth1965} constructed a further three families of finite semifields which are also two dimensional over a weak nucleus. Moreover, in the same article, Knuth describes operations that allow one to obtain up to six semifields from a given semifield. We show how these operations in fact relate these four families of finite semifields and that up to isotopy there are two families, one which generates two non-isotopic semifields under the Knuth operations and the other which generates three non-isotopic semifields