The radical of the four generated algebra of alternating type

We consider the algebra over a field $K$ with a set of
generators $a_{1},a_{2},\ldots , a_{n}$ and defined by homogeneous
relations of the form
$a_{1}a_{2}\cdots a_{n} =a_{\sigma (1)} a_{\sigma (2)} \cdots a_{\sigma (n)}$,
where $\sigma$ runs through $\Alt_{n}$, the alternating group of
degree $n$. It is shown that for $n=4$ the algebra is
semiprimitive provided $\mbox{char} (K)\neq 2$. If $\mbox{char}
(K)=2$, then it is proved that the Jacobson radical is a finitely
generated ideal that is nilpotent and it is determined by a
congruence on the underlying monoid, defined by the same
presentation. Such a result was proved in an earlier paper for
$n\geq 5$. The proof for $n=4$ is more complicated.