How can I recover the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids? I am using the formulation of the theorem given in

Rui Loja Fernandes, Lie Algebroids, Holonomy and Characteristic Classes, Advances in Mathematics 170 (2002) pp 119–179, doi:10.1006/aima.2001.2070, arXiv:math/0007132.

On Page 13, Fernades gives the following hint:

In general, the structure functions that appear in relations (11) will depend both on the $x$'s and $y$'s variables, subject to (12). For special classes of Lie algebroids one might have extra information that leads to further simplification of the structure functions. For example, in the case of a Poisson manifold, one always has the relationship: $$ c^{ij}_k = \frac{\partial b^{ij}}{\partial x^k}. $$ Then, all structure functions in (11) depend only on the $y$'s variables, and one obtains the Weinstein Splitting Theorem.

I made some fruitless attempts =(


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