A variety $V$ of modal algebras is called tabular if it is generated by a finite algebra. It is called pretabular if it is not tabular, but every proper subvariety is tabular. [1]
Claim: Suppose that $V$ is pretabular. Then, every infinite finitely generated algebra in $V$ generates $V$.
Proof: Suppose that $A \in V$ is infinite and finitely generated, and let $W$ be the subvariety of $V$ generated by $A$. If $W$ is proper, then it is generated by a finite algebra, and therefore, it is locally finite, see e.g. [2], Theorem II.10.16. Since $A \in W$ and $A$ is finitely generated, it must by finite, contradicting the hypothesis. It follows that $A$ generates $V$. QED
[1] Blok, W. J. (1980). Pretabular varieties of modal algebras. Studia Logica: An International Journal for Symbolic Logic, 39(2):101–124.
[2] Burris, S. and Sankappanavar, H. P. (2012). A Course in Universal Algebra - The Millenium Edition, 2012 Update. Springer Verlag, New York.
This is quite surprising to me (and somehow hard to believe), so I wonder if the proof is correct. and, if not, where it fails.