Separating SiO2 and CaSO4 electrostatically.  A mixture of SiO2 and CaSO4 could also be melted along with some other additions to make glass.  The CaSO4 would decompose to CaO.  An excellent article by G. Landiss Glassmaking on the Moon
BELOW) Magma or molten silicate electrolysis can be used to obtain oxygen, ceramic bricks and ferrosilicon at early Moonbases.  Ceramic bricks can be used to build heavy retorts for solar silicothermic & carbothermic reduction and ferrosilicon can be mixed with iron from illmenite reduction to make high silicon alloy iron that has high corrision resistance for making heavy sulfuric acid leaching vats and plumbing that must handle acid.  It's always better to use in situ resources.  High silicon alloy iron is called Duriron.  read more
BELOW)  Zone refining can purify metals and concentrate trace elements.  It's easer to extract trace elements from more concentrated material than from dilute moodust.  If we refine KREEPy regolith we can concentrate valuable rare earths, uranium and thorium which will be vital to lunar industry.  Unless we find good astroblemes where Ni, Cu, Zn, Au, Pt, etc. rich irony asteroids impacted, we must work with elements found in moondust.  Getting at those traces and producing new alloys with available elements instead copper and zinc alloys of aluminum and magnesium that are common on Earth is of importance. Rare earth elements from KREEP might be used for alloying. Steels made with Cr, Mn, Si and other lunar available elements rather than nickel is another goal we have. Geologists now say that the nickel and other metals mined at the Sudbury astrobleme in Ontario, Canada upwelled from deep in the Earth. If so, why aren't other volcanic upwellings rich in PGMs, nickel, copper, etc.? I am enough to an eccentric to believe that many of the chalcophiles at Sudbury came from an asteroid that was once a part of the deep mantle or outer core where chalcophiles sink to of a long shattered body that gave rise to the Main Belt asteroids.  A dreamer, yes. An expert, no. An asteroid impact crater rich in chalcophiles and siderophiles would be of immense value to lunar industry, but things don't exist just because we wish them too...
Silicon rod after boiling of Si+Al from magma electrolysis or solar carbothermic reduction contains traces of Al, Cr, Mn
Electrostatic separation may have more uses.  The sulfates produced by the sulfuric acid leach might be separated electrostatically, or they might be calcined with solar heat to oxides that are then separated electrostatically.  If this is possbile, it would then be easier to reduce MgO with silicon, boil off the magnesium and condense it in a ceramic retort.  The Al2O3 might be reduced with carbon and solar heat, or it might be carbochlorinated in titanium chambers (Ti resists attack by Cl) and electrolyzed to get pure aluminum.  Electrolysis of AlCl3 does not consume the electrodes nor does it require cyrolite which will be hard to get on the Moon. However, due to the low boiling point of AlCl3 temps. must be kept down and pressure must be applied.  Small percentages of MnO2, CrO, Cr2O3 and sodium oxide might also be separated electrostatically from calcined sulfates and reduced with heat+carbon or carbochlorinated and electrolyzed.  It might even be possible to directly electrolyze these oxides in FFC cells.  Much research remains to be done.  These pages only offer possible routes to lunar element separation from regolith. 
                                             Electrostatic Separation

Different materials will absorb static electric charge at different voltages and polarities.  This makes it possible to separate them. For example, you can remove the hulls from ground coffee beans with a comb that has been rubbed on wool to give it a static electric charge.  Industry uses electrostatic separation to sort minerals, salts, food products, plastics and other materials. Electrostatic separation will work well on the Moon with no interfering air currents or humidity.  No water or other solvents in short supply on the Moon are necessary. Temperature and particle size of the materials extracted from regolith must be controlled.  The effects of low lunar gravity will cause materials to behave differently than on Earth.  Experimentation in orbital centrifuges that simulate lunar gravity or at a research base on the Moon with actual lunar regolith is necessary. 

Materials will either be repelled or depressed, held on to, by a static electric charge. Non-reversible minerals are repelled by a positive or a negative electric charge.  Two non-reversible materials can be separated at a given voltage, negative or positive, because one will be repelled further from the electrode than the other.    Reversible materials are either reversible positive (RP) or reversible negative (RN).  Reversible positive materials are repelled by a positive charge and RN materials are depressed by a positive charge.  Reversible negative materials are repelled by a negative charge and RP materials depressed.
We can see that SiO2 and CaSO4 could be easily separated. With a positive electrode voltage of  7800V CaSO4 will be repelled but not SiO2.  Separating other compounds  is certainly possible but this would require more data and some experimentation.  A manned and/or robotic Moon base where real world experiments are done before tackling the job of mining thousands even millions of tons of regolith and extracting metals for lunar power beaming systems or solar power satellites and other construction jobs would be wise.  It would be necessary to determine the electrostatic behavior of actual lunar materials (which will not be pure and will vary slightly in composition at different locations) in lunar conditions (gravity, temperatures, vacuum, etc.) and not just regolith simulant in labs on Earth to get accurate data.
Information from Mining Engineer.com   and  Mining Engineer.com fig.1
CaSO4 repelled by (+) charge
       MISCELLANEOUS IDEAS
Electrostatic separation could be used to get various minerals in relatively pure form before chemical treatment.  Actual experiments with Moon dust are required.
Data for electrostatic separation is based on separations done in air.  We must do experiment in vacuum chambers with real lunar soil sample from Apollo missions.  Once we know the right voltages for various lunar minerals, we can calculate the arcs for repelled minerals in low lunar gravity and design and build an electrostatic separation machine for work on the Moon.