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We study the tangent and cotangent bundles of a Lie group G which are also Lie groups. Our main results are to show that on TG the canonical Jacobi endomorphism field S is parallel with respect to the canonical Lie group connection Lie group and that dually on the cotangent bundle of G the canonical symplectic form is parallel with respect to the canonical connection. We next prove some theorems for Lie algebra extensions in which we can obtain a group representation for the extended algebra from the representation of the lower dimensional algebra. We also determine the Lie algebra of the automorphism group of three well known Lie algebras. Finally we study the Hamilton-Jacobi separability of conformally flat metrics and find a metric, Lagrangian and geodesics for the solvable codimension one nilradical six dimensional Lie Algebras where one exists.