In pointless topology, for any locale $A$ (a.k.a. complete Heyting algebra) its Heyting negation (a.k.a. pseudocompelement) $\neg : A \to A$ gives rise to a nucleus $\neg\neg: A \to A$. As is also well known, the associated sublocale $A_{\neg\neg} = \{x \in A: \neg\neg x = x\}$ happens to be Boolean.
Is this sublocale the largest of all the Boolean sublocales of $A$? If not, does $A_{\neg\neg}$ have a special property among its kind?
Here are related facts of which I am aware:
$A_{\neg\neg}$ is the least sublocale (that is, least of all, possibly non-Boolean sublocales) containing the zero of $A$.
The Boolean sublocales of $A$ are exactly the sublocales determined by nuclei of the form $((\cdot)\to a) \to a$ for $a \in A$. (Write $b(a)$ for such a sublocale.) For instance, $A_{\neg\neg} = b(0)$.
Nonetheless, it is not necessarily the case that $b(a) \subseteq b(0)$ for all $a \in A$. This is essentially because $x \to a \equiv ((x \to a) \to 0) \to 0$ is not a law of Heyting algebras.
My references for those facts are Stone Spaces by Johnstone and Frames and Locales by Picado and Pultr.