Here's what Brienne wrote about Jaime in the Game of Thrones finale. There was a lot to resolve in the show's ending, but perhaps the most low-key and emotional was the new knight honoring her fallen lover.
The Book of Brothers is a log of all the Kingsguard, with a list of all their heroic and legendary deeds. It was previously seen in the Game of Thrones season 4 premiere when Joffery mocked his uncle/father for such a short list of deeds. This is a slight by both the keeper of the book and the King at how Jaime's life is defined by his Kingslayer persona. By the time of the show's ending, he's had a couple of extra lines added that take in the reign of Tommen:
Squired for Barristan Selmy against the Kingswood outlaws. Knighted and named to the Kingsguard in his sixteenth year for valor in the field. At the sack of King's Landing, murdered his King, Aerys, at the foot of the Iron Throne:
Pardoned by King Robert Baratheon:
Thereafter known as the Kingslayer:
After the murder of King Joffrey I by Tyrion Lannister, served under King Tommen I:
However, in the Game of Thrones finale, Brienne corrects that, honoring Jaime's claim that "there's still time" to rack up some real achievements. She writes everything he's done during the show, cementing him as a hero and no longer defined by an action made decades ago:
Captured in the field in the Whispering Wood:
Set free by Lady Catelyn Stark in return for an oath to find and [protect] her two daughters:
Lost his [hand.]
Took Riverrun from the Tully rebels, without loss of life.
Lured the Unsullied into ruling Casterly Rock, sacrificing his childhood home in service for a greater strategy.
Outwitted the Targaryen forces to seize Highgarden. Fought at the Battle of the Goldroad bravely, narrowly escaping death by dragonfire.
Pledged himself to the forces of men and rode north to join them at Winterfell, alone.
Faced the Army of the Dead, and defended the castle against impossible odds until the defeat of the Night King. Escaped imprisonment and rode south in an attempt to save the capital from destruction.
Died protecting his Queen.
Back in Game of Thrones season 1, Jaime was captured by Robb Stark during the Northern rebellion before being freed by Catelyn to find and protect Sansa and Arya. He lost his hand in season 3. The rest of the events are more recent: taking Riverrun with only threats not actions was in season 6; tricking the Unsullied and Targaryens was season 7; and his stand against the White Walkers and death with Cersei are this final season. While these were morally complex actions, they nevertheless chart Jaime's redemption - even protecting Cersei is one of earnest love.
What's notably absent is any mention of Brienne, one of the most important characters in Jaime's arc. She escorted him to King's Landing, looked after him after he lost his hand, carried forward the oath to help the Stark girls, and eventually had a night of passion with him in Winterfell. Not mentioning herself is a selfless move to not impose herself on Jaime's story, but with the final line also a tragic admittance that he really did love Cersei.
This skewed telling of past events is reflected in Sam's A Song of Ice and Fire book, which apparently features no mention of key player Tyrion Lannister. It would seem, in its final episode, Game of Thrones is making a point about how the small and personal can still be lost with time, and how that isn't necessarily a bad thing. Indeed, both Brienne and Tyrion are members of King Bran's new small council (Head of the Kingsguard and Hand of the King respectively); they will be part of new legends. Further, though, considering how Game of Thrones season 8 has been criticized for its fast pace, this seems like a rather open admission by the showrunners to the nature of storytelling.