BA is British Airways. BAE Systems is the new name for what used to be British Aerospace, which used BAe as its abbreviation, to avoid confusion with BA.

I don't know why it bothers me, but I really dislike the used-to-be-an-acronym-but-not-any-more names. For example SRI (not affiliated with Stanford so the S doesn't mean that any more), and my least favourite "HRL Laboratories" which we have to pretend does not expand to "Hughes Research Laboratories Laboratories". Yuk.