  We develop the structure theory of symplectic Lie groups based on the study
of their isotropic normal subgroups. The article consists of three main parts.
In the first part we show that every symplectic Lie group admits a sequence of
subsequent symplectic reductions to a unique irreducible symplectic Lie group.
The second part concerns the symplectic geometry of cotangent symplectic Lie
groups and the theory of Lagrangian extensions of flat Lie groups. In the third
part of the article we analyze the existence problem for Lagrangian normal
subgroups in nilpotent symplectic Lie groups.
