Further info:

For all practical purposes it would be impossible for the ancients to find elemental aluminium. Acc. to Wikipedia it’s only found in some cold seeps, and amounts should be minuscule. It might not look like it, but this stuff is damn reactive (cue to thermite).

Initially it was produced already from bauxite, but the Deville-Pechiney and Deville used for that were fairly complex and costly, that’s why the resulting aluminium was pricier than gold. In a nutshell:

1. [DP] Roast bauxite, soda, and charcoal at 1200°C, to get sodium aluminate. Use good charcoal, or the gunk in it will get down the production chain.
2. [DP] Dissolve the aluminate with sodium hydroxide.
3. [DP] Bubble the solution so it precipitates aluminium hydroxide.
4. [DP] Roast the precipitate to get aluminium oxide.
5. React the alumina with hydrochloric acid, to get hydrated aluminium chloride. Then dry it the best you can.
6. [D] Roast soda and charcoal alone this time. You’ll get metallic sodium.
7. [D] React the metallic sodium with the hydrated aluminium chloride. Note a lot of sodium will be consumed by the hydration water.

In the meantime, the Bayer and Hall-Héroult processes used nowadays are way simpler:

1. [B] Mix bauxite with sodium hydroxide, under pressure at 180°C, so you get a solution. Discard the red mud*.
2. [B] Cool it down and add a bit of water. The pH will lower so the aluminium hydroxide will precipitate, and if you use seed crystals it’ll be mostly pure. Discard the solution*.
3. [B] Roast the precipitate until it dries out. I’m not sure but I think the leftover liquid can be re-concentrated for reuse in step #1.
4. [HH] Mix the precipitate with cryolite and electrolyse it. The cryolite can be mostly reused, as it’s a catalyst.

*I’m saying “discard” but both the red mud and the solution can be further processed, specially if the bauxite in question contains titanium and/or gallium.