The Goddess of Flash

Shihouin Yoruichi, the exiled daughter to another one of the Four Noble Houses, appears to be one of Byakuya's oldest acquaintances--or rival, in the captain's view. Considering their equality in noble status they were bound to befriend each other, though Yoruichi is significantly older.

By the present storyline there is no more room for their relationship to grow, but I find it to be well-fleshed out in the little historical context we are given for it. It also helps reinforce Byakuya's subconscious desire for freedom.

The Tease

Yoruichi Byakuya Yoruichi

Though Yoruichi is no less capable in ability and prowess as a shinigami, she has managed to maintain an upbeat attitude through her noble years and onward. In Turn Back the Pendulum she playfully goads Byakuya into chasing her after stealing his hairtie.

The hopeful side of me believes that Yoruichi mentored the boy not only to improve his skills in shunpo ("Flash Step"), but to show him that there is still fun to be had in spite of noble responsibilities.

The One Who Left

You can't catch me with that level of shunpo.

Though the Bleach timeline gets a little tricky, it's estimated that Yoruichi was absent from Soul Society for at least 100 years before returning to mentor Ichigo. In all that time, based on her brief interaction with him, Byakuya never forgot their senpai-kouhai relationship. He proudly proclaims that he's trained all this time and can surpass her, but just as with their games of tag, she manages to elude him in the end.

The fact that Byakuya's temperament doesn't miss a beat when his old friend/rival/mentor shows up after about a century indicates that his feelings about her haven't changed much. He acts loftier than her because he thought that not only was he better than her skill-wise, but also that her abandonment of country, title, and status were deplorable.

In reality though, I think part of Byakuya was envious that Yoruichi could be so laid-back, and also that she could flee Soul Society when the situation got desperate. He never admits it, but no one would blame him if he felt shackled by the very law he was meant to uphold. When considering that, his hostility toward Yoruichi may have been a twisted kind of admiration, one that he felt for the better part of his life.