Today (2016) Aluminium is still only economically extractable from minerals that do not contain silicon. Silicon is the second most common element in earth's crust just after oxygen.
As waste copious amounts or red mud  are produced. Red mud is mostly composed out of iron oxides, titanium oxides and alkaline silicic acid compounds. The problematic part is that red mud also contains ~1% soluble heavy metal salts (V Cr As) which makes it rather toxic. Advanced atomically precise technology should make it possible to fully extract those (potentially valuable) heavy metals directly by efficient chemomechanic processing and indirectly by the availability of very cheap energy. This is a form of atomically precise disassembly which is harder than mechanosynthesis and thus not to expect early on.
The most obvious form to use aluminium in advanced atomically precise technology is in the form of the naturally occuring gemstone leuco-sapphire (aka colorless gem grade corundum) other strongly metastable polymorphs though may be of equal or even higher interest due to their different crystal structure.
Hexagonal polymorphs are:
- α-Al2O3 leuco-sapphire (Mohs 9 | Trigonal).
- χ-Al2O3 ( Mohs ?? | hexagonal)
Cubic polymorphs (may be of special interest due to their high symmetry):
- γ-Al2O3 (known to be strongly metastable) (Mohs 8)
(Is there no natural mineral with that structure?)
Monoclinic (maybe less interesting due to very low symmetry):
(TODO: Identify the exact structure types of the polymorphs with Wyckoff points such that related compounds with other elements replacing aluminium can be identified. Those can then be used to explore the corresponding pseudo phase diagrams and to induce isostructural bending)
Limitedness of stable binary compounds
Most other nonmetal compounds of aluminium beside Al2O3 strongly react with water and thus may not be considered as useful structural building materials.
- aluminium nitride AlN (surface hydrolysis creates a passivation layer on macroscopic chunks) this may still be useful
- aluminium carbide Al4C3 (reacts with water to aluminium hydroxide and methane - irritant)
- aluminium fluoride AlF3 (water soluble; interestingly low toxicity)
- aluminium sulfide Al2S3 (toxic due to hydrogen sulfide H2S generation when contacting water)
- aluminium phosphide AlP (highly toxic due to generation of phosphine PH3 when contacting water)
- aluminium chloride AlCl3 (water soluble; toxic neurotoxine)
Mechanosynthetisized nonthermodynamic checkerboard pattern compounds like the following may be stable:
- aluminium sulfoxide Al4O3</sub>S3
- aluminium phosphonitride Al2NP
More aluminium based gemstone materials
A good point to start are always the natural aluminium minerals . Since they need to be insoluble to persist on the surface (except in very dry environments).
There are many aluminium silicate minerals  like:
- Al2SiO5 Andalusite  & Sillimanite  (Mohs 6.5-7)
- Al6Si2O13 Mullite  (Mohs 6-7)
- Al2SiO5 Kyanite  (Mohs 4.5-7 anisotropic)
Salts of aluminium usually have low solubility in water. Albeit aluminiums extreme abundance in our environment (most abundant metal in earth's crust) it does not play a known role in human biology. Elevated intake of aluminium salts (rare in nature) is suspected of having detrimental effects on health.