Why is it precisely the Lie bracket that encodes the information about a given transformation?
A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given transformation. Of course there are different representations for each Lie group/algebra, but all representations describe the same transformation, for the different objects they act on.
Is there any way to gain insight into why the bracket makes sure we end up with the correct transformation?
To elaborate on what I have in mind: One can start with a given transformation, acting on a given type of object. For example rotations (here described by the usual rotation matrices) on vectors. Following the standard procedures one ends up with the Lie algebra for the rotation group. Now one can throw away everything one knows about rotations of vectors (i.e. the rotation matrices) and start using only the Lie algebra. Again following standard procedures one can construct the irreducible representations of this group from the Lie algbra. The vector representation one started with can then be seen as a special case. Another representation for example would be the representation describing rotations of spinors.
Why is the Lie bracket enough to encode what the group does?
PS: I understand that the bracket is important, as can be seen for example in the Baker–Campbell–Hausdorff formula. Nevertheless I do not understand why the bracket is the defining feature.