Editing Quaternion (section)

=== Hamilton product ===
For two elements {{math|''a''<sub>1</sub> + ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}} and {{math|''a''<sub>2</sub> + ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}}, their product, called the  '''Hamilton product''' ({{math|''a''<sub>1</sub> + ''b''<sub>1</sub>'''i''' + ''c''<sub>1</sub>'''j''' + ''d''<sub>1</sub>'''k'''}}) ({{math|''a''<sub>2</sub> + ''b''<sub>2</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''d''<sub>2</sub>'''k'''}}), is determined by the products of the basis elements and the [[distributive law]]. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

<math display=block>\begin{alignat}{4}
        &a_1a_2  &&+ a_1b_2 \mathbf i   &&+ a_1c_2 \mathbf j   &&+ a_1d_2 \mathbf k\\
  {}+{} &b_1a_2 \mathbf i &&+ b_1b_2 \mathbf i^2 &&+ b_1c_2 \mathbf{ij} &&+ b_1d_2 \mathbf{ik}\\
  {}+{} &c_1a_2 \mathbf j &&+ c_1b_2 \mathbf{ji} &&+ c_1c_2 \mathbf j^2 &&+ c_1d_2 \mathbf{jk}\\
  {}+{} &d_1a_2 \mathbf k &&+ d_1b_2 \mathbf{ki} &&+ d_1c_2 \mathbf{kj}  &&+ d_1d_2 \mathbf k^2
\end{alignat}</math>

Now the basis elements can be multiplied using the rules given above to get:<ref name="SeeHazewinkel" />

<math display=block>\begin{alignat}{4}
          &a_1a_2 &&- b_1b_2 &&- c_1c_2 &&- d_1d_2\\
   {}+{} (&a_1b_2 &&+ b_1a_2 &&+ c_1d_2 &&- d_1c_2) \mathbf i\\
   {}+{} (&a_1c_2 &&- b_1d_2 &&+ c_1a_2 &&+ d_1b_2) \mathbf j\\
   {}+{} (&a_1d_2 &&+ b_1c_2 &&- c_1b_2 &&+ d_1a_2) \mathbf k
\end{alignat}</math>