The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that $H$ is a representation of some Lie group $G$. So you want to understand this Lie group $G$, and generally you do it by looking at its Lie algebra. At least initially, this is understood as a way to make the problem of classifying Lie groups and their representations easier.

But there are Lie algebras which are not the Lie algebra of a Lie group, and people are still interested in them. One possible way to justify this perspective from a physical point of view is that Lie algebras might still be viewable as ("infinitesimal"?) symmetries of physical systems in some sense. However, I have never seen a precise statement of how this works (and maybe I just haven't read carefully enough). Can anyone enlighten me?