To prove that $(G, o)$ is an abelian group, we need to verify that the set $G = \{ q \in \mathbb{Q} \mid q \neq -1 \}$ with the binary operation $o$ defined by $x \, o \, y = x + y + xy$ satisfies the four properties of an abelian group:

1. Closure: $G$ is closed under the operation $o$.
2. Associativity: The operation $o$ is associative.
3. Identity element: There exists an identity element $e \in G$ such that $x \, o \, e = e \, o \, x = x$ for all $x \in G$.
4. Inverse element: For each element (x