  In this work we extend the Lu-Weinstein construction of double symplectic
groupoids to any Lie bialgebroid such that its associated Courant algebroid is
transitive and its Atiyah algebroid integrable. We illustrate this result by
showing how it generalises many of the examples of double symplectic groupoids
that have appeared in the literature. As preliminary steps for this
construction, we give a classification of exact twisted Courant algebroids over
Lie groupoids (CA-groupoids for short) and we show the existence of a foliation
by twisted Courant algebroids on the base of a twisted CA-groupoid.
