I asked this here https://math.stackexchange.com/questions/1441408/orbit-under-lie-group-and-projective-variety but did not get any answers, so I am asking here. I apologize if this is not appropriate for this site.
On https://en.wikipedia.org/wiki/Generalized_flag_variety#Highest_weight_orbits_and_homogeneous_projective_varieties there is a section which says
If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way.
I am interested in the case of Lie groups. Does anyone have a reference for this, which includes a proof? The bold parts are what really interest me.
I am also interested in whether the action of the real form $G_0 \subset G$ on the highest weight vector is also a projective algebraic variety.