  When ${\frak g}$ is a complex semisimple Lie algebra, we study the variety
${\mathcal L}$ of subalgebras of ${\frak g}\oplus{\frak g}$ that are maximally
isotropic with respect to $K_1 - K_2$, where $K_i$ is the Killing form on the
ith factor. We show the irreducible components of ${\mathcal L}$ are smooth,
classify them in terms of the generalized Belavin-Drinfeld triples introduced
by Schiffmann, and relate them to orbits of the adjoint group $G\times G$.
Building on ideas of Yakimov, we give a new proof of Karolinsky's
classification of the diagonal $G$-orbits in ${\mathcal L}$. Our proof enables
us to compute of the normalizer in ${\frak g}$ of a subalgebra in ${\mathcal
L}$ under the diagonal action. As a consequence, we recover the classification
of Belavin-Drinfeld triples. By results of math.DG/9909005, ${\mathcal L}$ is a
Poisson variety and we determine the rank of the symplectic leaf at each point
of ${\mathcal L}$ in terms of combinatorial data and relate the symplectic
leaves to intersections of orbits of subgroups of $G\times G$. As a
consequence, an intrinsically defined Poisson structure on each conjugacy class
on $G$ has an open symplectic leaf and we determine the rank at each point of
the conjugacy class.
