Quasigroup is not necessarily B-Algebra

Theorem
Let $\left({S, \circ}\right)$ be a quasigroup.

Then $\left({S, \circ}\right)$ is not necessarily a $B$-algebra.

Proof
As all groups are quasigroups we will use a small group as a counterexample.

Consider the Cayley table of the group of order 3:


 * $\begin{array}{c|cccccc}

& 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \\ \end{array}$

By inspection we see that $B$-algebra axiom $(A2)$ does not hold as $1 \circ 1 \ne 0$.

Also see

 * $B$-Algebra is a Quasigroup, where it is shown that all $B$-algebras are quasigroups.