Editing Octonion (section)

===Matrix representation===
Just as quaternions can be [[Quaternion#Matrix_representations|represented as matrices]], octonions can be represented as tables of quaternions. Specifically, because any octonion can be defined a pair of quaternions, we represent the octonion <math> ( q_0, q_1 )</math> as:
<math display=block>\begin{bmatrix}
         q_0 & q_1 \\
        -q_1^* & q_0^*
\end{bmatrix}</math>

Using a slightly modified (non-associative) quaternionic matrix multiplication:
<math display=block>\begin{bmatrix}
         \alpha_0 & \alpha_1 \\
        \alpha_2 & \alpha_3
\end{bmatrix}\circ\begin{bmatrix}
         \beta_0 & \beta_1 \\
        \beta_2 & \beta_3
\end{bmatrix}=\begin{bmatrix}
         \alpha_0\beta_0+\beta_2\alpha_1 & \beta_1\alpha_0+\alpha_1\beta_3\\
        \beta_0\alpha_2+\alpha_3\beta_2 & \alpha_2\beta_1+\alpha_3\beta_3
\end{bmatrix}</math>
we can translate octonion addition and multiplication to the respective operations on quaternionic matrices.<ref name="Ensembles"></ref>