a_3  $

  $\nabla \cdot \textbf{A}$     $-(\partial_1 A_1 + \partial_2 A_2 + \partial_3 A_3) \emph{\textbf{I}}_0 $

  $\nabla \times \textbf{A}$    $-\emph{\textbf{I}}_1 ( \partial_2
                                                                 A_3 - \partial_3 A_2 ) - \emph{\textbf{I}}_2 ( \partial_3 A_1
                                                                 - \partial_1 A_3 ) - \emph{\textbf{I}}_3 ( \partial_1 A_2
                                                                 - \partial_2 A_1 )$

  $\nabla \circ \textbf{A}_0$   $\emph{\textbf{I}}_1 \partial_1 A_0
                                                                 + \emph{\textbf{I}}_2 \partial_2 A_0
                                                                 + \emph{\textbf{I}}_3 \partial_3 A_0  $

  $\partial_0 \textbf{A}$       $\emph{\textbf{I}}_1 \partial_0 A_1
                                                                 + \emph{\textbf{I}}_2 \partial_0 A_2
                                                                 + \emph{\textbf{I}}_3 \partial_0 A_3  $
  ------------------------------------------------------------------------------------------------------------------------------

  : The multiplication of the operator and octonion physics quantity.[]{data-label="tab:table2"}

\[sec:level2\]Electromagnetic field equations
---------------------------------------------

In the electromagnetic theory described with the complex quaternion, it is able to deduce directly the Maxwell’s equations in the classical electromagnetic theory. In this approach, substituting the $S$-quaternion for the quaternion, one can obtain the same conclusions still.

In the $\emph{S}$-quaternion space $\mathbb{H}_e = ( \emph{\textbf{I}}_0, \emph{\textbf{I}}_1, \emph{\textbf{I}}_2, \emph{\textbf{I}}_3 )$, the electromagnetic potential is $\mathbb{A}_e = \emph{i} \textbf{A}_0 + \textbf{A}$, the electromagnetic strength is $\mathbb{F}_e = \square \circ \mathbb{A}_e$. The definition of electromagnetic strength can be expressed as $$\begin{aligned}
\mathbb{F}_e = ( - \partial_0 \textbf{A}_0 + \nabla \cdot \textbf{A} ) + \emph{i} ( \partial_0 \textbf{A} + \nabla \circ \textbf{A}_0 ) + \nabla \times \textbf{A} ~,\end{aligned}$$ where the vector potential of electromagnetic 