Abelian ideals and the variety of Lagrangian subalgebras

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