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The coupling of the tangent bundle $TM$ with the Lie algebra bundle $L$(K.Mackenzie,2005, Definition 7.2.2) plays the crucial role in theclassification of the transitive Lie algebroids for Lie algebra bundle $L$ withfixed finite dimensional Lie algebra $\rg$ as a fiber of $L$. Here we give anecessary and sufficient condition for existence such a coupling. Namely wedefine a new topology on the group $\Aut(\rg)$ of all automorphisms of Liealgebra $\rg$ and show that tangent bundle $TM$ can be coupled with the Liealgebra bundle $L$ if and only if the Lie algebra bundle L admits a localtrivial structure with structural group endowed with such new topology. Thesecond version of the article is different from the first because of theimprovement of the text by invaluable efforts of professor J.Stasheff, to whomwe submit our sincere gratitude.

The Existence of Coupling in the Category of Transitive Lie Algebroid