On the integration of transitive Lie algebroids

We revisit the problem of integrating Lie algebroids A double right arrow M to Lie groupoids G paired right arrows M, for the special case that the Lie algebroid A is transitive. We obtain a geometric explanation of the Crainic-Fernandes obstructions for this situation, and an explicit construction of the integration whenever these obstructions vanish. We also indicate an extension of this approach to regular Lie algebroids.