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.