Commutative B-Algebra Induces Abelian Group

Theorem
Let $\left({X, \circ }\right)$ be a commutative $B$-algebra.

Let $*$ be the binary operation on $X$ defined as:


 * $\forall a, b \in X: a * b := a \circ \left({0 \circ b}\right)$

Then the algebraic structure $\left({X, *}\right)$ is an abelian group such that:


 * $\forall x \in X: 0 \circ x$ is the inverse element of $x$ under $*$.

That is:
 * $\forall a, b \in X: a * b^{-1} := a \circ b$

Proof
From B-Algebra Induces Group, the algebraic structure $\left({X, *}\right)$ is a group such that:


 * $\forall x \in X: 0 \circ x$ is the inverse element of $x$ under $*$.

It remains to show that $*$ is a commutative operation.

Let $x, y \in X$:

Hence the result.

Also see

 * Abelian Group Induces Commutative $B$-Algebra