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 group axioms in turn:

G0: Closure
Let: be quaternions.
 * $$\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$$
 * $$\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$$

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 quaaternion $$\mathbf x$$ in the form of a matrix:
 * $$\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$$

From Properties of Matrix Entrywise Addition, matrix 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 the commutativity of real numbers.