What I always miss from this introductory abridged explanations, and what makes the connection between Lie groups and algebras ('infinitesimal' groups) really useful, is that the exponential process is a universal mechanism, and provides a natural way to find representations and operators (eg Lie commutator, the BCH formula) where the group elements can be transformed through algebraic manipulations and vice-versa. That discovery offers a unified treatment of concepts in number theory, differential geometry, operator theory, quantum theory and beyond.