Is there any way to refine Feldspar?

Feldspars are mostly composed of three compounds:

• KAlSi$_3$O$_8$

• NaAlSi$_3$O$_8$

• CaAl$_2$Si$_2$O$_8$

I notice they each have Aluminum. Feldspar is far more common than Bauxite (the only Aluminum ore commercially mined). As it stands now, Feldspar is used in glass-making, but I'm wondering if there's some chemical refining process that can extract the Aluminum somehow?---especially that third one, since it has two Al's.

Note: I'm not looking for an economical method. That's proly asking too much. I'm not worried about aluminum reserves, because bauxite is still very plentiful. I'm simply interested in a chemical method to extract Aluminum, or at least Alumina (Al$_2$O$_3$), from Feldspar, purely for reasons of chemical curiosity.

The only thing I've researched is the Bayer Process, but I have no idea if it can be applied or modified for Feldspar.

Aluminium oxide can be isolated from feldspar by first acid digestion with a mixture of hydrofluoric acid and sulfuric acid (get rid of silicon by formation of silicon tetrafluoride), then digesting the residue with sodium-potassium carbonate, dissolving the obtained mass in dilute hydrochloric acid and then precipitating $\ce{Al(OH)3}$ by adding ammonia. The $\ce{Al(OH)3}$ can be calcined to aluminium oxide by heating to over 1100°C.