\mathbb{A}_e$. The gauge conditions of field potential are chosen as $f_0 = 0$ and $F_0 = 0$. The operator is $ \square = \emph{i} \emph{\textbf{i}}_0 \partial_0 + \Sigma \emph{\textbf{i}}_k \partial_k$. $ \nabla = \Sigma \emph{\textbf{i}}_k \partial_k$, $ \partial_j = \partial / \partial r_j$. $v_0$ is the speed of light.

The octonion field source $\mathbb{S}$ of the electromagnetic and gravitational fields can be defined as, $$\begin{aligned}
\mu \mathbb{S} && = - ( \emph{i} \mathbb{F} / v_0 + \square )^* \circ  \mathbb{F}
\nonumber \\
&& = \mu_g \mathbb{S}_g + k_{eg} \mu_e \mathbb{S}_e - ( \emph{i} \mathbb{F} / v_0 )^* \circ \mathbb{F} ~,\end{aligned}$$ where $\mu$, $\mu_g$, and $\mu_e$ are coefficients. $\mu_g < 0$, and $\mu_e > 0$. $*$ denotes the conjugation of octonion. In general, the contribution of the tiny term, $( \emph{i} \mathbb{F}^* \circ \mathbb{F}/ v_0 )$, in the above could be neglected.

According to the coefficient $k_{eg}$ and the basis vectors, $\mathbb{H}_g$ and $\mathbb{H}_e$, the octonion definition of the field source can be separated into two parts, $$\begin{aligned}
&& \mu_g \mathbb{S}_g = - \square^* \circ \mathbb{F}_g  ~,
\\
&& \mu_e \mathbb{S}_e = - \square^* \circ \mathbb{F}_e  ~,\end{aligned}$$ where Eq.(3) is the definition of gravitational source, while Eq.(4) is the definition of 