A tribble can be seen and heard purring on Captain Lorca's desk in his first scene. The tribbles were introduced in The Trouble with Tribbles (1967) and later appeared in More Tribbles, More Troubles (1973), Star Trek III: The Search for Spock (1984), Trials and Tribble-ations (1996), The Breach (2003) and Star Trek Beyond (2016).
Executive Producer Aaron Harberts explained that the episode title "Context is for Kings" related to "trying to see what Starfleet and the Federation are up against in this particular context" and the idea that Lorca was "a captain who really almost could only exist in this particular context," with Starfleet at war.
For the scenes aboard the USS Glenn, the set selected for filming was also used as the interiors of the Shenzhou and the Discovery, with just a few modifications to make the Glenn itself seem more ruined.
The code that Stamets gives Burnham to fix has been identified by some as being Microsoft Windows coding, specifically decompiled code for the Stuxnet virus.
Captain Lorca's old ship was the USS Buran. Buran was the name of the Soviet's space shuttle. Lorca's new ship is the USS Discovery. Discovery was the 3rd American space shuttle. The Soviet Buran made 1 unmanned trip to space and never flew again.