A Lie group is a smooth manifold whose tangent space at its origin is its Lie algebra. Taking an example for lie group such as SL(2), and due to above facts we should then be able to translate the abstract study of manifolds to some concrete computations dealing only with numbers. What are some references that exactly do that?
Just to give some examples, we know beforehand that tangent space will contain matrices with so and so property for some particular Lie group. And then we compute to show exactly that. Similarly, can we concretely define geodesics on Lie groups totally in a computational manner?
References are fine. In case, there are certain abstract manifold concepts that cannot be computationally translated in terms of Lie group, please mention them and say some lines as to why. A philosophical answer is also sufficient.