WARNING: Spoilers for Cobra Kai season 3.
Cobra Kai season 3 explains John Kreese's (Martin Kove) origin story as well as what inspired the creation of the Cobra Kai dojo. During several flashback sequences, the Netflix series details Kreese's service during the Vietnam War and the context for his martial arts training. By the Cobra Kai season 3 finale, "December 19," the true origin story for Kreese's dojo is revealed.
Kreese appears in the original Karate Kid trilogy as the primary baddie. Importantly, however, the movies don't fully account for the character's "no mercy" mentality, yet there are occasional references to his military background. In The Karate Kid Part III, Kreese enlists a military colleague named Terry Silver (Thomas Ian Griffith) to help intimidate his karate rivals Daniel LaRusso (Ralph Macchio) and Mr. Miyagi (Pat Morita). Three decades later in Cobra Kai, Johnny Lawrence (William Zabka) re-establishes the titular dojo and learns that his former sensei is not only alive but has been living in homeless shelters. Kreese claims that he'd worked as a military instructor, but it's heavily implied that he's suffering from Post-Traumatic Stress Syndrome. By the end of Cobra Kai season 2, Kreese reassumes control of his dojo, primarily because of Johnny's perceived weaknesses as a leader.
A flashback in Cobra Kai season 3 reveals that Kreese learned martial arts while training for a special task force unit during the Vietnam War. Specifically, he learns the Korean style of Tang Soo Do from his commanding officer, Captain Turner (Terry Serpico), who previously studied under Master Kim Sun-Yung during the Korean War. In The Karate Kid, Kreese teaches Tang Soo Do to his Cobra Kai students, which contrasts with the Miyagi-Do method taught by Mr. Miyagi; a style that was inspired by the real-life Okinawan practices kata and Gōjū-ryū. In addition, the fight sequences in The Karate Kid were choreographed by Pat E. Johnson, who learned Tang Soo Do from both Kang Do Hee and Chuck Norris. In Cobra Kai, Johnny and Kreese's students are trained in the style of Tang Soo Do, which makes their methods not only unique but also distinct from their rivals.
Cobra Kai season 3 also reveals that Kreese's "no mercy" mantra aligns with the moment he was recruited by Captain Turner. In The Karate Kid franchise, Mr. Miyagi's peaceful practices differ with the aggressive style of Cobra Kai, which ultimately becomes a major source of conflict in the Netflix series. For example, Kreese only re-assumes control of his dojo when Johnny eases up on his "no mercy" teachings, which unfortunately leads to a massive school brawl. The flashback sequences in Cobra Kai season 3 reinforce Kreese's core values, and shows that they were learned from Captain Turner: strength, determination, and brutality; no hesitation, no second thoughts, and - most importantly - no mercy.
In an ironic twist, a death match between Kreese and Captain Turner in Vietnam ultimately inspires the name of the Cobra Kai dojo. The special task force unit is captured by Viet Cong soldiers, who make their prisoners fight over a cobra pit. Crucially, one of Kreese's comrades is named "Silver," who is rumored to be none other than the future financier of the Cobra Kai dojo, Terry Silver, as revealed in The Karate Kid Part III. In Cobra Kai, Silver is initially supposed to fight against Captain Turner but Kreese ultimately takes his place and utilizes all the lessons learned from his superior officer to defeat him. First, he uses combat skills. Then, Kreese shows "no mercy" by kicking Captain Turner into the cobra pit, even though U.S. soldiers have just arrived to save the POWs. In the present, Kreese remains serious about preserving the values of Cobra Kai because he literally once killed a comrade who urged him to be "more than just a soldier."