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The main aim in this book is to look for TCOA(triple co- action of reductive Lie group)with new properties on Lie group from a lemma of Schure,the literature are concerned with studying the action of Lie algebra of two representations,one is usuall and the other is dual,while our interest in this work is focused on some action on Lie group.A new action of tensor product and direct sum for representations of Lie groups has been defind.The theoretical justifications are developed and proved supported by some concluding remarks and illustrations.