  We define a general notion of abstract double Lie algebroid. We show (1) that
the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in
this sense; (2) that the double cotangent constructed from Lie algebroid
structures on a vector bundle A and its dual A* is a double Lie algebroid if
and only if (A, A*) is a Lie bialgebroid; (3) that a vacant double Lie
algebroid structure is equivalent to a matched pair structure on the side Lie
algebroids.
