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This lecture covers the characterization of tori and diagonalizable groups, focusing on the properties of diagonalizable groups and their equivalence with connectedness and torsion-freeness. It also discusses the image of diagonalizable subgroups under homomorphisms and their regularity. Moreover, it explores the conjugacy of diagonalizable subgroups and their commutativity, as well as the density of elements of finite order in diagonalizable subgroups.