The system obtained by adding full propositional quantification to S5 is known to be decidable, while that obtained by doing so for T is known to be recursively intertranslatable with full second-order logic. Recently it was shown that the system with two S5 operators and full propositional quantification is also recursively intertranslatable with second-order logic. This note establishes that the map assigning p to \squarep provides a validity and satisfaction preserving translation between the T system and the double S5 system, thus providing an easier proof of the recent result.
"A Simple Embedding of T into Double S5." Notre Dame J. Formal Logic 45 (1) 13 - 18, 2004. https://doi.org/10.1305/ndjfl/1094155276