  For a semisimple algebraic group $G$ of adjoint type with Lie algebra
$\mathfrak g$ over the complex numbers, we establish a bijection between the
set of closed orbits of the group $G \ltimes \mathfrak g^{\ast}$ acting on the
variety of Lagrangian subalgebras of $\mathfrak g \ltimes \mathfrak g^{\ast}$
and the set of abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. In
particular, the number of such orbits equals $2^{\text{rk} \mathfrak g}$ by
Peterson's theorem on abelian ideals.
