Looking at some filters generated by elements of a finite Boolean algebra I have the impression that many/most/all of them are by themselves Boolean algebras (at least I didn't stumble upon a counterexample). For example the filter generated by $\lbrace 1,2\rbrace$ and $\lbrace 1,3\rbrace$ in the power set algebra of $\lbrace 1,2,3,4\rbrace$ is isomorphic to the power set algebra of $\lbrace 1,2,3\rbrace$ - with $\lbrace 1\rbrace$ instead of $\emptyset$ being the minimal element and accordingly $\lbrace 1,3,4\rbrace$ instead of $\lbrace 3,4\rbrace$ being the negation of $\lbrace 1,2\rbrace$.
My question is:
Which filters of which Boolean algebras are by themselves Boolean algebras?
If the answer is not "all filters of all Boolean algebras" what is the smallest/simplest counterexample?