The variety $\SBA$ of skew Boolean algebras, introduced by Leech in~\cite{Leec90}, is a natural example of a binary discriminator variety. Central to the study of binary discriminator varieties is the variety $\iBCS$ of implicative BCS-algebras, first considered by the authors in~\cite{Bign02a}. In~\cite{Bign02a}, it is shown that $\iBCS$ is generated (as a variety) by a certain three-element algebra~$\BTwo$, initially investigated by Blok and Raftery in~\cite{Blok95}. In the first part of this paper, we show that the quasivariety $\QBTwo$ generated by~$\BTwo$ is the class of all $\zseq{\bs, 0}$-subreducts of $\SBA$. Using insights from the theory of skew Boolean algebras, we investigate $\QBTwo$ in the second part of this paper, obtaining a fairly complete elementary theory. In particular, we characterise $\QBTwo$ as a subclass of $\iBCS$; provide a finite axiomatisation of $\QBTwo$; describe the $\QBTwo$-subdirectly irreducible algebras; and characterise the lattice of subquasivarieties of $\QBTwo$. Collectively, the results may be understood as a generalisation to the `non-commutative' situation of several well known theorems of classical algebraic logic connecting implicative BCK-algebras with (generalised) Boolean algebras.