  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 the
classification of the transitive Lie algebroids for Lie algebra bundle $L$ with
fixed finite dimensional Lie algebra $\rg$ as a fiber of $L$. Here we give a
necessary and sufficient condition for existence such a coupling. Namely we
define a new topology on the group $\Aut(\rg)$ of all automorphisms of Lie
algebra $\rg$ and show that tangent bundle $TM$ can be coupled with the Lie
algebra bundle $L$ if and only if the Lie algebra bundle L admits a local
trivial structure with structural group endowed with such new topology. The
second version of the article is different from the first because of the
improvement of the text by invaluable efforts of professor J.Stasheff, to whom
we submit our sincere gratitude.
