Comment by FBT

> ... Does distributivity of inversion ~ over operation ⋆ follow from the other Abelian group axioms / properties? If so, how?

It does. For all x and y:

  (1) ~x ⋆ x = 0 (definition of the inverse)
  (2) ~y ⋆ y = 0 (definition of the inverse)
  (3) (~x ⋆ x) ⋆ (~y ⋆ y) = 0 ⋆ 0 = 0 (from (1) and (2))
  (4) (~x ⋆ ~y) ⋆ (x ⋆ y) = 0 (via associativity and commutativity)

In (4) we see that (~x ⋆ ~y) is the inverse of (x ⋆ y). That is to say, ~(x ⋆ y) = (~x ⋆ ~y). QED.