Quaternion Multiplication

Theorem
Let $$\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$$ and $$\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$$ be quaternions.

Then their product is given by:

$$ $$ $$ $$

Proof
From Matrix Form of Quaternion we have that:
 * $$\mathbf x_1 \mathbf x_2 = \begin{bmatrix} a_1 + b_1 i & c_1 + d_1 i \\ -c_1 + d_1 i & a_1 - b_1 i \end{bmatrix} \begin{bmatrix} a_2 + b_2 i & c_2 + d_2 i \\ -c_2 + d_2 i & a_2 - b_2 i \end{bmatrix}$$

Let $$\mathbf x_1 \mathbf x_2 = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{bmatrix} = \begin{bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end{bmatrix}$$.

Throughout we use the definition of conventional matrix product.

So:

$$ $$ $$ $$

$$ $$ $$ $$

$$ $$ $$ $$

$$ $$ $$ $$

Hence the result from Matrix Form of Quaternion.