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Basic aluminium (orbasic aluminum) is the name of more than one functional group consisting of aluminium with one or two hydroxy groups attached.
Dihydroxyaluminium, Al(OH)2, also known as dibasic aluminium, is monovalent, and known in these compounds:
Hydroxyaluminium, Al(OH), also known as monobasic aluminiumorbasic aluminium, is divalent, and known in these compounds:
Aluminium, Al, is trivalent. Aluminium triacetate, Al(CH3CO2)3, is a complete molecule without any hydroxy groups, so it is not a "basic aluminium" compound.
Aluminium hydroxide, Al(OH)3, aluminium with three hydroxy groups attached, is a complete molecule, so it is not a "basic aluminium" compound.
Aluminium acetate is a name for three salts in the solid state: dihydroxyaluminium aluminium acetate, hydroxyaluminium diacetate, and aluminium triacetate, Al(CH3CO2)3. In aqueous solution, aluminium triacetate hydrolyses to form a mixture of the other two, so all solutions of all three can be referred to simply as "aluminium acetate", as the species co-exist and inter-convert in chemical equilibrium.
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