I am doing a project on Lie groups and Lie algebras, and am hoping to include a proof of Lie's Third Theorem using Ado's Theorem. I have been following the proof of Ado's Theorem in Structure and Geometry of Lie Groups, which applies to all finite-dimensional Lie algebras (I assume it means over any field of characteristic zero?).
I have now realised that this proof requires a huge amount of prerequisite structure theory of Lie algebras. If I restrict the scope of my project to real, finite-dimensional Lie groups, is there a shorter/more accessible proof that I could find somewhere? The problem is that this project starts with the absolute fundamentals of Lie theory and I may not be able to include Ado's Theorem if there is not a more direct approach I can find, since there are quite strict page limits.
Also, will the 'restriction of faithful finite-dimensional representation to nilradical is nilpotent' part of the proof be necessary for a proof of Lie's Third Theorem?
Thanks in advance
Best Answer
Knapp's book Lie Groups: Beyond an Introduction includes a proof of Ado's theorem restricted to precisely this case in an appendix. There is no easy way around some structure theory but everything that is needed cited precisely and included in the main body of the book.