Quaternions Defined by Ordered Pairs

Theorem
Consider the quaternions $\Bbb H$ as numbers in the form:
 * $a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$

where:
 * $a, b, c, d$ are real numbers;


 * $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are entities related to each other in the following way:

Now consider the quaternions $\Bbb H$ defined as ordered pairs $\tuple {x, y}$ where $x, y \in \C$ are complex numbers, on which the operation of multiplication is defined as follows:

Let $w = a_1 + b_1 i, x = c_1 + d_1 i, y = a_2 + b_2 i, z = c_2 + d_2 i$ be complex numbers.

Then $\tuple {w, x} \tuple {y, z}$ is defined as:


 * $\tuple {w, x} \tuple {y, z} := \tuple {w y - z \overline x, \overline w z + x y}$

where $\overline w$ and $\overline x$ are the complex conjugates of $w$ and $x$ respectively.

These two definitions are equivalent.

Proof
First we identify the following:

We can see that:

Let:
 * $w = a_1 + b_1 i$
 * $x = d_1 + c_1 i$
 * $y = a_2 + b_2 i$
 * $z = d_2 + c_2 i$

and so:
 * $\overline w = a_1 - b_1 i$
 * $\overline x = d_1 - c_1 i$

Then substituting for $(1)$ to $(4)$ above, we have:

Notice the way $\mathbf j$ and $\mathbf k$ are configured. See that they are what appears to be in the wrong order.

We can then demonstrate the equivalence by showing that:

which is equivalent to:

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

in accordance with Quaternion Multiplication.

Thus:

So:

as required.