I'm sorry, but this seems ridiculous. Aluminum is obtained from naturally ocurring Aluminum oxides. We are told upthread that Gallium is "mostly obtained as a byproduct of aluminum production_. In other words:

• you mine ores rich in Aluminum oxide
  
• you extract metallic Aluminum from it (and Gallium as a by-product) by means of an energy-intensive process
  
• you then recombine the Aluminum and Gallium to obtain Aluminum oxide and Hydrogen

There is no free lunch here, might as well use the electricity spent producing the Aluminum for ordinary electrolysis of water.

By the way, the reason Gallium is an impurity in Aluminum oxides is that Aluminum and Gallium are chemically similar [both contiguous elements in the the Earth Metal (s2p1) series, Al is 1s2 2s2p6 3s2p1 and Gallium is 1s2 2s2p6 3s2p6 4s2d10p1].

Bush is a symptom, not the disease.