I have been using the concepts of Lie group, Lie algebra and some of its properties for quite a while now in various geometry courses, but I had to pick those concepts along the way because they are always taken for granted. I want to fill this hole and I was wondering which (graduate) book will be best suited for this, some research led me to this list, although I am open for any suggestion:
-Brian C. Hall, Lie groups, Lie algebras and representations. Currently my least favorite option, mainly because of the answer given here. I do know differential geometry and I would like to study this subject in all generality. Still people seem to like this book, and it has a lot of problems, which I appreciate very much.
-V. S. Varadarajan, Lie groups, Lie algebras and their representations. I know very little about this book, I have been told that it is kind of a more general version of the last one (using differential geometry freely), but I did not manage to find a copy on any library to check it out. Does it have a good amount of problems?
-J. Harris and W. Fulton, Representation theory. Seems to be a classic, but (and I am judging just by the title for I could not get a copy of this one either) I am afraid this book will go too fast on the basics of Lie groups and Lie algebras. Also, how about the problems on this one?
Hall's book is excellent. You can't go wrong there.
I would also suggest supplementing with Chapter 4 of Tu's book for more of a complete connection with the geometry (Hall's book largely focuses on the representation theory of Lie Groups and Lie Algebras, although there is geometry in that too in later chapters).