  We study Lagrangian subalgebras of a semisimple Lie algebra with respect to
the imaginary part of the Killing form. We show that the variety $\Lagr$ of
Lagrangian subalgebras carries a natural Poisson structure $\Pi$. We determine
the irreducible components of $\Lagr$, and we show that each irreducible
component is a smooth fiber bundle over a generalized flag variety, and that
the fiber is the product of the real points of a De Concini-Procesi
compactification and a compact homogeneous space. We study some properties of
the Poisson structure $\Pi$ and show that it contains many interesting Poisson
submanifolds.
