Quaternion Addition forms Abelian Group

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

Then $\left({\mathbb H, +}\right)$, where $+$ denotes quaternion addition, is an abelian group.

Proof
Taking the abelian group axioms in turn:

G0: Closure
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:
 * $\mathbf x_1 + \mathbf x_2 = \left({a_1 + a_2}\right) \mathbf 1 + \left({b_1 + b_2}\right) \mathbf i + \left({c_1 + c_2}\right) \mathbf j + \left({d_1 + d_2}\right) \mathbf k$

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 $\left({a_1 + a_2}\right) \mathbf 1 + \left({b_1 + b_2}\right) \mathbf i + \left({c_1 + c_2}\right) \mathbf j + \left({d_1 + d_2}\right) \mathbf k$ is a quaternion.

Hence $\left({\mathbb H, +}\right)$ is closed.

G1: Associativity
From Matrix Form of Quaternion, we can express a quaternion $\mathbf x$ in the form of a matrix:
 * $\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$

We have that Matrix Entrywise Addition is Associative.

It follows that quaternion addition is also associative

G2: Identity
The identity element of $\left({\mathbb H, +}\right)$ is:
 * $\mathbf 0 = 0 \mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$

as can be seen:

G3: Inverses
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

The inverse of $\left({\mathbb H, +}\right)$ 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.