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?