Jaime has been recovering his honour ever since the middle of season 3, and it culminates in the events of this episode. Cersei asks Jaime if he would find Sansa and kill her. Old Jaime would have done that without hesitation, but new Jaime just can't bring himself to do it, instead tasking Brienne not just with finding Sansa, but with getting her safely away from Cersei's grasp. He entrusts her with his new Valyrian sword, which she appropriately names Oathkeeper, and sends her off. Tagging along as her squire is Pod, who clearly relishes the chance to loyally serve another honourable person. This is a huge moment for Jaime because it shows how much he has changed since we first saw him. He wants to restore his honour, and to do that he feels he must keep the oath he swore to Catelyn Stark. Since he can't do that given the position he is in, he sends Brienne off to perform the deed for him as a sign of loyalty and trust. The symbol of this trust is the sword Oathkeeper; the giving away of which is incredibly selfless on Jaime's part. Tywin wouldn't be pleased. The performances from Nikolaj Coster-Waldau and Gwendoline Christie are fantastically nuanced; able to convey an array of emotions just through body language and eye contact, their goodbye is the sweetest moment of the episode, especially given that we don't know if they will ever meet again.
