Montgomery and Zippin "Topological Transformation Groups" (originally published in 1955) is still the only book to cover the relevant results on topological characterization of Lie groups in full generality (including Lie group actions). I am not sure if this belongs to algebra or topology area-wise, but it is used in my area, geometric group theory.
For pedagogical purposes, I still use "What Is Mathematics?" by Courant and Robbins (originally published in 1941) and "Geometry and Imagination" (1932) by Hilbert and Kohn-Vossen, when a high school student or an undergraduate asks me for suggestions.
My personal definition of an "old book" is the same as Lee Mosher's, so I do not include here Chapters 4-6 of Bourbaki's "Lie groups and Lie algebras" (1968) which I use as a working tool.