@Torre I have some additional questions please if you can kindly answer me. I want to ask about the display mode, for example for the lie algebra when I execute in contrast from your code it shows me only the nonzero brackets. and the tangent vectors are in the form D_x instead of (partial derivative)

as in your execution code. how to change this. I want to ask also about the DifferentialGeometry package available under the name DifferentialGeometryUSU.mla on Utah State University website. Is it the exact same package that came by default with maple 2015 ? if not, do I have to install this one ?

@Torre This really helps me a lot. Thank you very much for all the time you spent writing the answer.