Jaime Lannister (Nikolaj Coster-Waldau) is one of the most complex characters throughout the A Song of Ice and Fire franchise. The dishonorable knight in love with his sister, who, in the very first episode of Game of Thrones pushes a 10-year-old boy out the window with the intent to kill him, became a fan favorite.
Jaime starts his run in the series as a seemingly shallow, one-note villain. Labeled "the Kingslayer," he became infamous throughout Westeros for betraying his oath as a member of the Kingsguard when he murdered Aerys Targaryen, the Mad King. However, as the story progresses, Jaime becomes a viewpoint character, adding far more depth than most fans would have initially expected.
Jaime's journey throughout the later seasons of Game of Thrones sees him torn between two different loves, his sister Cersei Lannister (Lena Headey) and a knight named Brienne of Tarth (Gwendoline Christie). Interestingly, these two characters share something very specific in common with Jaime, adding extra nuance to his connection with both.
Jaime Lannister: The Kingslayer
While bathing in Harrenhal with Brienne in season 3, Jaime confesses his true motivations for killing Aerys. Most people did not know the extent to which the Mad King had gone mad. According to Jaime, he had planted Wildfire caches beneath the city and had planned to detonate them, which would cause mass casualties throughout King's Landing. After Aerys declared "Burn them all," Jaime killed him to protect the civilians who now hate him for saving their lives because of an oath he broke to do so.
After telling Brienne the truth, Jaime nearly passes out. Brienne catches him, shouting "Kingslayer!" in concern. Jaime then replies, "Jaime. My name is Jaime." Here, Jaime formally rejects his Kingslayer title. His greatest act of heroism has become his greatest source of shame.

Jaime Lannister is not the only Kingslayer
Several other characters throughout A Song of Ice and Fire have killed kings. Notably, this includes both Cersei and Brienne, who each play a part in the death of an important ruler.
Cersei helps to orchestrate the death of King Robert Baratheon (Mark Addy), her abusive husband. While on a hunt, she instructs her cousin, Lancel Lannister (Eugene Simon), to pour Robert too much wine, getting him as drunk as possible, In his stupor, Robert suffers grievous wounds while attempting to hunt a boar, ultimately dying upon his return to King's Landing. This allows Cersei's son, Joffrey (Jack Gleeson), to ascend the throne, and frees Cersei from Robert's wrath.
The difference between Cersei's Kingslaying and Jaime's is that Cersei's role in Robert's death is far subtler, allowing her scheming to remain undetected until the High Sparrow (Jonathan Pryce) wrings a confession out of Lancel in season 5. While Jaime must suffer the common folk's taunts and japes for killing Aerys, Cersei manages to avoid a similar fate for some time.
Meanwhile, Brienne is first accused of killing her king, Renly Baratheon (Gethin Anthony), since she stood by him at the time of his death. While audiences know the true killers to be Stannis Baratheon (Stephen Dillane) and Melisandre (Carice van Houten), who created a magical demon to sneak into Renly's camp undetected, Brienne ends up taking the fall for Renly's death, resulting in her receiving similar pariah status as Jaime.
Later, Brienne actually becomes a Kingslayer when she finds a wounded Stannis and kills him, finally avenging Renly. In contrast to Jaime, Brienne is proud to become a Kingslayer, since Stannis had killed the man to whom she swore an oath, making her "Kingslaying" journey the inverse of his.

Even more Kingslayers
Debatably, Cersei and Brienne are not the only Kingslaying characters close to Jaime. His brother, Tyrion (Peter Dinklage) and father, Tywin (Charles Dance) may also fit the bill.
Tyrion is falsely accused of killing his nephew, King Joffrey. After Oberyn Martell (Pedro Pascal), Tyrion's champion, loses his trial by combat, he is found guilty, resulting in his escape from Westeros. Although Tyrion never actually kills a king, he is still marred with the Kingslayer nickname, much like his brother.
Tywin, meanwhile, helps orchestrate the infamous Red Wedding, resulting in the death of King Robb Stark (Richard Madden). While Tywin himself is not present for the massacre, his stamp is all over, with the wedding band breaking into his song, "The Rains of Castamere," when the killing begins. Roose Bolton (Michael McElhatton), the man who actually kills Robb, declares, "The Lannisters send their regards" as he plunges the knife into his chest, confirming Tywin's involvement to Robb in his final moments.
While Tywin never receives the "Kingslayer" title, he is as much responsible for the death of a king as his eldest son.
Arguably, these are the four most important relationships for Jaime throughout the series. His ties to Cersei and Brienne define his character's journey, while he truly loves his brother and looks up to his father. Perhaps the shared Kingslaying connection subconsciously brings Jaime closer to Cersei, Brienne, Tyrion, and Tywin, allowing each of them to understand and empathize with Jaime (except for the famously un-empathetic Tywin) in ways most other characters can't.
For more from the world of Westeros, make sure to stream House of the Dragon and A Knight of the Seven Kingdoms on HBO Max.
