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We study integrable systems on double Lie algebras in absence of Ad-invariantbilinear form by passing to the semidirect product with the $\tau$-representation. We show that in this stage a natural Ad-invariant bilinearform does exist, allowing for a straightforward application of the AKS theory,and giving rise to Manin triple structure, thus bringing the problem to therealm of Lie bialgebras and Poisson-Lie groups.

Double Lie algebras, semidirect product, and integrable systems