I myself am working on two theorems on the Lie transformation group from the book Kobayashi Transformation Group in Differential Geometry

sentenceTo let $ mathfrak {S} $ the group of the differentiable transformation of the manifold $ M $ and $ mathcal {S} $ Let be the set of all vector fields $ X in mathfrak {X} (M) $ this creates 1 parameter subgroup $ varphi_ {t} = text {exp} (tX) $

the transformation for $ M $ so that $ varphi_ {t} in mathfrak {S} $, The sentence $ mathcal {S} $ Use parentheses of the vector field to define a Lie algebra. Then, when $ mathcal {S} $ is the finite-dimensional Lie algebra of vector fields $ M $ then $ mathfrak {S} $ Lie group of transformation and $ mathcal {S} $ his lie algebra.

The idea of the proof is quite simple, take the Lie algebra $ mathfrak {g} ^ {*} $ generated by $ mathcal {S} $, if $ mathfrak {g} ^ {*} $ if finite-dimensional, then according to the third set of Lie, there are simply connected Lie-groups $ mathfrak {S} ^ {*} $ the last can be chosen so that $ mathfrak {S} ^ {*} subset mathfrak {S} $ Using local action etc … more can be shown that it is a normally connected subgroup and is open. At the moment, everything seems normal, but then the author claims that the smooth structure can be converted into another connected component or simply agrees $ mathfrak {S} $ if I'm not mistaken, I can not get it here, unless we accept the left translation or the right one for $ g in mathfrak {S} $, the assignment $ L_ {g}: mathfrak {S} ^ {*} longrightarrow g. Mathfrak {S} ^ {*} $ be differentiable.

My question is about the idea of how we transfer smooth structure from connected normal subgroups?

There is an essay by Richard S.Plais. A global formulation of the lie theory of transformation groups contains a lot of things about it, but I could not see exactly where the answer is.