I’ve jumped ahead a bit in my reread to reread the “good” bits of A Feast for Crows (that is to say, I like the entire novel, but I was really interested in focusing on the Jaime, Cersei and Brienne chapters), and specifically in talking about Jaime’s last chapter where he throws Cersei’s letter onto the fire.
From the beginning of Jaime’s POV, Martin has juxtaposed Jaime, Cersei and Brienne for us, and the burning of Cersei’s letter is another of these instances, (though not as immediately obvious as it was with Jaime’s weirwood dream in which Cersei takes away the “only light in the world” and leaves him in darkness but then Brienne lights that darkness with her glowing blue sword - Oathkeeper - or maybe Lightbringer?)
Still, the parallels between Jaime’s being sent away from Brienne by Roose Bolton, and returning for her when she is in mortal danger and Jaime’s being sent away from Cersei by Cersei and refusing to return for her when she begs are completely intentional.
Jaime leaves Brienne to an uncertain fate to return to Cersei, and then dreams of her, as a beauty, as a knight, and as his champion against the vengeful ghosts of Rhaegar and his Kingsguard brethren. Because of his dream, he returns and intervenes in a sort of mock “trial by combat” between Brienne and the bear. (“Mock” only because the intent was to mock Brienne AND because there’s no chance that she’s coming out alive - her innocence or guilt are not at stake because she has committed no crime and because Hoat doesn’t care about justice in any form anyway.)
Unlike Cersei, Brienne doesn’t ask for help; Jaime provides it of his own free will (once he’s ascertained that she cannot help herself which is one of the things I love so much about the bear pit … Jaime has every faith that given the right weapon, Brienne could defeat the bear.) Jaime’s intervention is spectacularly Jaime-like, showing the same reckless courage that drove him to try and fight his way to Robb at the Whispering Wood and the same impulsiveness that led him to throw Bran from the tower (in other words, the same qualities that have always shaped Jaime are used here for a good purpose - saving Brienne - rather than for a terrible one.)
Jaime has no plan when he leaps into the bear pit other than to put his body between Brienne and the bear. We’re inside his head, so we know that at the moment when he makes that wild jump, he’s hasn’t calculated that Steelshanks Walton fears Roose Bolton enough that he’ll save Jaime and by default, Brienne as well. His immediate impulse is to throw himself into danger alongside Brienne, to die with her if he can’t save her. (“If you want her, go get her.” So he did.)
In contrast, Jaime doesn’t seem particularly interested in dying for or with Cersei by the end of Feast. We’ve seen time and again that Jaime cannot fight with his left hand so what Cersei is asking of him is, in fact, to die (and then she will also die presumably.) He thinks back on his decision in Dance and there’s a cold rationality to his thoughts that never comes up with the bearpit. He can’t answer the question of why he went back for Brienne, but he knows very well why he didn’t go back for Cersei. “Even if he had gone back, he could not hope to save her. She was guilty of every treason laid against her, and he was short a sword hand.”
Some of Jaime’s indifference stems from his belief that Cersei has made her own bed (ignoring good advice from both Kevan and Jaime, trusting the untrustworthy Aurane Waters and Littlefinger), some of it is sexual jealousy (the toxic refrain of Lancel, and Kettleblack and Moonboy), and a lot of it is finally recognizing a deep truth about their relationship: he always loved her far more single-mindedly than she loved him and that a great deal of what he loved was an illusion that he created in order to justify the terrible things he did on behalf of that love. That is to say, I do think Cersei loves Jaime (or at any rate, the Jaime she sees in her head - the beautiful boy who is the best swordsman in Westeros and who does her bidding, rather than the aging, crippled man who has begun to think for himself), but she also loves a lot of other people/things. In her own emotionally-crippled-by-being-Tywin-Lannister’s-child way, Cersei loves her children, and of course, she loves power and being Queen, and we know that she would have happily married Rhaegar; Jaime, in contrast, loved Cersei (and Tyrion) and that was basically it.
The other truth about Jaime’s and Cersei’s relationship - which Jaime also has slowly come to recognize - is that their love is built on a false image that each has of the other, an image created and shaped in huge part by their father. It’s no coincidence that the last time they have sex is in front of Joffrey’s body, when sex is the only thing with which they can still reach other because neither Jaime nor Cersei can comprehend the depth of loss that the other is feeling, making a mockery of Cersei’s belief that Jaime is “her other half.” Even as they are physically reunited in that scene, they are emotionally worlds apart. Just before Jaime burns Cersei’s letter, he dreams of his mother, who weeps when she says that Tywin wanted Jaime to be a great knight, Cersei to be a queen, and both to be so strong and brave and beautiful that no one would ever laugh at them, when the reality is that Jaime is a cripple, and Cersei is a terrible queen, and both of them have or will be mercilessly mocked. Jaime experienced this with the Bloody Mummers; Cersei will experience it during her Walk of Shame (and in both instances, as much as we, the readers, might have wanted both Jaime and Cersei to be “punished” for their evil deeds, the punishment is so unsatisfying and so unjust - as it is not for the actual crimes they’ve committed - that leaves a terrible taste in our mouths.)
Throughout Feast, Jaime’s POVs swing between the world-view of the man who loved Cersei and that of the man who rediscovered something of himself while he was with Brienne. In fact, more than a few times, Jaime’s chapters are physically sandwiched between those of Cersei and Brienne so while some of this is just an artifact of splitting Feast/Dance into two books, it’s also a very interesting contrast between the two women: Cersei driven close to madness by her paranoia (in some ways justified, in others not) versus Brienne, driven inexorably to the terrible choice that Stoneheart gives her; Cersei, who wants to be Queen but who has no sense of what it means to rule or that she has any responsibility to the ruled versus Brienne, who lives the suffering that the commonfolk endure and sees the price of a ruined country (and who, as best she can, being one lone woman, does try to put things right); Cersei, whose response to the intense misogyny of the society she lives in is to internalize that misogyny, who gives women to Qyburn so that he can create a monster for her, versus Brienne who risks her own life in a hopeless battle to save an inn full of children from monsters who rape and murder children.
And then there’s Jaime, who begins to shift from one side of the scale to the other, morally speaking. Jaime, who is torn between his love for Cersei and his still-intact physical desire for her and his willingness to do anything for her and for that love, versus the Jaime who once wanted to be Ser Arthur Dayne. Brienne is not Jaime’s “path to redemption” or his salvation (only Jaime can save Jaime), but what she is for him is a mirror into what he once was - the boy he thinks died just as surely as Aerys did. Jaime’s first interactions with Brienne, before he is maimed, often involve his trying to break her idealism, trying to make her see the world as the wretched place that he believes it is (in much the same way that Sandor Clegane voices his contempt for the knighthood and its supposed ideals to Sansa Stark.) And then, gradually, because of Brienne, Jaime realizes that it actually IS possible to live in the world and to know it and to still try to be true to the core of yourself, because that’s what Brienne does.
And maybe that’s really all that’s left to you in the end - that lonely dark path where you keep trying to do the right thing even when everything’s been stripped away from you - Brienne faces that moment, both when she defends the children at the inn, and when she is brought before Stoneheart, and in a subtler way, Jaime faces it too, when he realizes that he gave up his honor for the false coin of Cersei’s love. Even his Aunt Genna thinks he’ll be the Kingslayer until he dies, and yet there’s a part of Jaime that won’t give up on being Goldenhand the Just, despite what everyone thinks. What a tragic irony that the only person who genuinely believes Jaime has changed (even more, perhaps than Jaime himself believes it!) has, I think, sworn an oath to kill Jaime.