The Johannes Brahms paraphrase that Spock plays was written especially for this episode by Ivan Ditmars.
One of many Star Trek productions resembling William Shakespeare's The Tempest and/or Irving Block's Forbidden Planet (1956).
Flint's viewscreen appears to be a fairly close predictor of the modern flat panel TV that would become a commonplace household device decades later, except in 4:3 rather than widescreen format.
The undercarriage of Flint's robot, M-4, is a reused portion from the upper carriage of Nomad from The Changeling (1967).
Mr. Flint invites Kirk, Spock and Bones to his palace, which is the same as seen from afar in The Cage (1966): an Eastern palace with blue details, golden rooftops, a giant moon or other planet in the background left and a smaller, Saturn-like planet left of that. In 'The Cage' this is the stage for the fight between Captain Pike and the quasi-Viking giant. In the remastered Star Trek, this image was replaced with a new background.