Aluminium is a reactive metal that is difficult to extract from ore, aluminium oxide (Al2O3). Direct reduction -- with carbon, for example -- is not economically viable since aluminium oxide has a melting point of about 2,000 °C. Therefore, it is extracted by electrolysis; that is, the aluminium oxide is dissolved in molten cryolite and then reduced to the pure metal. By this process, the operational temperature of the reduction cells is around 950 to 980 °C. Cryolite is found as a mineral in Greenland, but in industrial use it has been replaced by a synthetic substance. Cryolite is a mixture of aluminium, sodium, and calcium fluorides: (Na3AlF6). The aluminium oxide (a white powder) is obtained by refining bauxite in the Bayer process. (Previously, the Deville process was the predominant refining technology.)

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Recovery of the metal via recycling has become an important facet of the aluminium industry. Recycling involves melting the scrap, a process that uses only five percent of the energy needed to produce aluminium from ore.[8] Recycling was a low-profile activity until the late 1960s, when the growing use of aluminium beverage cans brought it to the public consciousness.

Electric power represents about 20% to 40% of the cost of producing aluminium, depending on the location of the smelter. Smelters tend to be situated where electric power is both plentiful and inexpensive, such as South Africa, the South Island of New Zealand, Australia, the People's Republic of China, the Middle East, Russia, Quebec and British Columbia in Canada, and Iceland.

I presume this 20%-40% is for the extraction of Aluminum from Bauxite ore, and that recycling aluminum takes only 5% of that. However, note that the proposed process will start from (presumably) recycled Aluminum and then produce Aluminum Oxide (so we go back toward Bauxite).

Most gallium is extracted from the crude aluminium hydroxide solution of the Bayer process for producing alumina and aluminum. A mercury cell electrolysis and hydrolysis of the amalgam with sodium hydroxide leads to sodium gallate. Electrolysis then gives gallium metal. For semiconductor use, further purification is carried out using zone melting, or else single crystal extraction from a melt (Czochralski process). Purities of 99.9999% are routinely achieved and commercially widely available.

As I said in a comment upthread, I don't believe this process is any better, energetically, than straight electrolysis of water.

Bush is a symptom, not the disease.