Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the identity form a Lie algebra, can we conclude that the group is a Lie group?
As an example of why I am asking, consider the ODE $y''=y'$. It has a $2-$parameter group of symmetry transformations given by $y\to \epsilon y$, and $y\to y+t$. It's not clear to me that this forms a Lie group since the transformations don't commute and I can't see a smooth structure. However, its generators, given by $\partial_y,y\partial_y$ form a Lie algebra.