electromagnetic source.

The octonion velocity $\mathbb{V}$ is defined as, $$\mathbb{V} = v_0 \partial_0 \mathbb{R}  ~,$$ where in the case for single one particle, a comparison with the classical electromagnetic (and gravitational) theory reveals that, $\mathbb{S}_g = m \mathbb{V}_g$, and $\mathbb{S}_e = q \mathbb{V}_e$. $m$ is the mass density, while $q$ is the density of electric charge.

            $ $                     $1$            $\emph{\textbf{i}}_1$    $\emph{\textbf{i}}_2$    $\emph{\textbf{i}}_3$    $\emph{\textbf{I}}_0$    $\emph{\textbf{I}}_1$    $\emph{\textbf{I}}_2$    $\emph{\textbf{I}}_3$
  ----------------------- ----------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
            $1$                     $1$            $\emph{\textbf{i}}_1$    $\emph{\textbf{i}}_2$    $\emph{\textbf{i}}_3$    $\emph{\textbf{I}}_0$    $\emph{\textbf{I}}_1$    $\emph{\textbf{I}}_2$    $\emph{\textbf{I}}_3$
   $\emph{\textbf{i}}_1$   $\emph{\textbf{i}}_1$            $-1$            $\emph{\textbf{i}}_3$    $-\emph{\textbf{i}}_2$   $\emph{\textbf{I}}_1$    $-\emph{\textbf{I}}_0$   $-\emph{\textbf{I}}_3$   $\emph{\textbf{I}}_2$
   $\emph{\textbf{i}}_2$   $\emph{\textbf{i}}_2$   $-\emph{\textbf{i}}_3$            $-1$            $\emph{\textbf{i}}_1$    $\emph{\textbf{I}}_2$    $\emph{\textbf{I}}_3$    $-\emph{\textbf{I}}_0$   $-\emph{\textbf{I}}_1$
   $\emph{\textbf{i}}_3$   $\emph{\textbf{i}}_3$   $\emph{\textbf{i}}_2$    $-\emph{\textbf{i}}_1$            $-1$            $\emph{\textbf{I}}_3$    $-\emph{\textbf{I}}_2$   $\emph{\textbf{I}}_1$    $-\emph{\textbf{I}}_0$
   $\emph{\textbf{I}}_0$   $\emph{\textbf{I}}_0$   $-\emph{\textbf{I}}_1$   $-\emph{\textbf{I}}_2$   $-\emph{\textbf{I}}_3$            $-1$            $\emph{\textbf{i}}_1$    $\emph{\textbf{i}}_2$    $\emph{\textbf{i}}_3$
   $\emph{\textbf{I}}_1$   $\emph{\textbf{I}}_1$   $\emph{\textbf{I}}_0$    $-\emph{\textbf{I}}_3$   $\emph{\textbf{I}}_2$    $-\emph{\textbf{i}}_1$            $-1$            $-\emph{\textbf{i}}_3$   $\emph{\textbf{i}}_2$
   $\emph{\textbf{I}}_2$   $\emph{\textbf{I}}_2$   $\emph{\textbf{I}}_3$    $\emph{\textbf{I}}_0$    $-\emph{\textbf{I}}_1$   $-\emph{\textbf{i}}_2$   $\emph{\textbf{i}}_3$             $-1$            $-\emph{\textbf{i}}_1$
   $\emph{\textbf{I}}_3$   $\emph{\textbf{I}}_3$   $-\emph{\textbf{I}}_2$   $\emph{\textbf{I}}_1$    $\emph{\textbf{I}}_0$    $-\emph{\textbf{i}}_3$   $-\emph{\textbf{i}}_2$   $\emph{\textbf{i}}_1$             $-1$

  : The multiplication table of octonion.[]{data-label="tab:table3"}

\[sec:level2\]Gravitational field equations
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In the quaternion space $\mathbb{H}_g = ( 1, \emph{\textbf{i}}_1, \emph{\textbf{i}}_2, \emph{\textbf{i}}_3 )$, the gravitational potential is $\mathbb{A}_g = \emph{i} a_0 + \textbf{a}$, the operator is $ \square = \emph{i} \partial_0 + \nabla $, the gravitational strength is $\mathbb{F}_g = \square \circ \mathbb{A}_g$. The 