Quaternion Addition forms Abelian Group

Theorem
Let $\mathbb H$ be the set of quaternions.

Then $\struct {\mathbb H, +}$, where $+$ denotes quaternion addition, is an abelian group.

Proof
Taking the abelian group axioms in turn:

Let:
 * $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$
 * $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$

be quaternions.

By definition of quaternion addition:
 * $\mathbf x_1 + \mathbf x_2 = \paren {a_1 + a_2} \mathbf 1 + \paren {b_1 + b_2} \mathbf i + \paren {c_1 + c_2} \mathbf j + \paren {d_1 + d_2} \mathbf k$

So as $a_1, a_2, b_1, b_2$ etc. are all elements of $\R$, then so are $a_1 + a_2, b_1 + b_2$ etc.

So $\paren {a_1 + a_2} \mathbf 1 + \paren {b_1 + b_2} \mathbf i + \paren {c_1 + c_2} \mathbf j + \paren {d_1 + d_2} \mathbf k$ is a quaternion.

Hence $\struct {\mathbb H, +}$ is closed.

From Matrix Form of Quaternion, we can express a quaternion $\mathbf x$ in the form of a matrix:
 * $\mathbf x = \begin {bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end {bmatrix}$

We have that Matrix Entrywise Addition is Associative.

It follows that quaternion addition is also associative

The identity element of $\struct \mathbb H, +}$ is:
 * $\mathbf 0 = 0 \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$

as can be seen:

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

The inverse of $\struct {\mathbb H, +}$ is:
 * $-\mathbf x = -a \mathbf 1 + -b \mathbf i + -c \mathbf j + -d \mathbf k$

as can be seen:

and similarly for $\mathbf x + \mathbf -x$.

Commutativity
Commutativity follows from Real Addition is Commutative.

Thus all the abelian group axioms are seen to be fulfilled.