  Let X be a projective hyperk\"ahler manifold containing a Lagrangian subtorus
L. We study intersections of deformations of L, defining a canonical almost
holomorphic map called L-reduction, which is not birational if and only if X
admits an almost holomorphic Lagrangian fibration with (strong) fibre L.
  In dimension four we prove that in the above situation there is always a
holomorphic Lagrangian fibration with fibre L, thus answering a question of
Beauville in this particular case.
