Lt. Charlene Masters does not wear her Lt. stripe.
In a shot of the viewscreen near the end of the episode you can see the planet but there's no stars in the background.
When Kirk escorts Lazarus to the bridge, there's a shot of McCoy as they leave and two Red Alert panels - one is flashing while the other one is burnt out.
The commodore says the distortion was felt in every quadrant of the galaxy and beyond. How does he know this? It takes weeks for messages to get back and forth (see "Balance of Terror") and how would he know what's going on outside the galaxy?
Despite the fact there's a warning of an alien invasion and Lazarus is at the center of things, not to mention kind of wacko, he wanders freely around the Enterprise.
Starfleet sends out a code factor 1 command - everyone is to go on alert status because of a possible alien invasion. Then the commodore asks Kirk's opinion on what's going on and Kirk says it's an alien invasion - duh!
The Lazarus beards keep getting thicker and thinner.
Spock: I fail to comprehend your indignation, sir. I have simply made the logical deduction that you are a liar.
Kirk: What was that?
Spock: What my instruments read is totally unbelievable, Captain. Twice, for a split second each time, everything within range of our instruments seemed on the verge of winking out.
Kirk: I want facts, not poetry.
Spock: Jim, madness has no purpose. Or reason. But it may have a goal.
Kirk: Sometimes pain can drive a man harder than pleasure.
Originally John Drew Barrymore was supposed to play Lazarus but he quit abruptly so Robert Brown was chosen to replace him.
The astronomical background shown when Lazarus is jumping between the parallel universes is the Trifid Nebula (also known as M20 and NGC 6514). From Earth, it is found in the region of the constellation Sagittarius.