Non-finitely related and finitely related monoids

We transform the method of Glasson into a sufficient condition under which a monoid is non-finitely related, add a new member to the collection of interlocking word-patterns, and use it to show that the monoid $M(ab^2a, a^2b^2)$ is non-finitely related. We also give a sufficient condition under which a monoid is finitely related and use it show that $M(a^2b^2)$ is finitely related. Together with the results of Glasson this completes the description of all finitely related monoids among the monoids of the form $M(W)$ where every word ${\bf u} \in W$ depends on two variables and every variable occurs twice in $\bf u$.