the basis vector $\emph{\textbf{I}}_0$ are the imaginary numbers, while the coordinates with the basis vector $\emph{\textbf{I}}_k$ are all real. In the paper they can be written as the preceding description form in the context, in order to deduce the Maxwell’s equations. Those restriction conditions are the indispensable components of the classical electromagnetic theory.

On the analogy of the coordinate definition of complex coordinate system, one can define the coordinate of octonion, which involves the quaternion and $\emph{S}$-quaternion simultaneously. In the octonion coordinate system, the octonion physics quantity can be defined as $ \{ ( \emph{i} c_0 + \emph{i} d_0 \textbf{\emph{I}}_0 ) \circ \emph{\textbf{i}}_0 + \Sigma ( c_k + d_k \textbf{\emph{I}}_0^* ) \circ \emph{\textbf{i}}_k \} $. It means that there are the quaternion coordinate $c_k$ and the $S$-quaternion coordinate $d_k \emph{\textbf{I}}_0^*$ for the basis vector $\emph{\textbf{i}}_k$, while the quaternion coordinate $c_0$ and the $S$-quaternion coordinate $d_0 \emph{\textbf{I}}_0$ for the basis vector $\emph{\textbf{i}}_0$. Herein $c_j$ and $d_j$ are all real.

\[sec:level1\]Equivalent Transformation
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\[sec:level2\]Gravitational Field Equations
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Making use of the transforming of the basis vectors of $\nabla$ in the quaternion operator $\square$, it is able to translate further the gravitational field equations, Eqs.(9)-(12), from the quaternion space $\mathbb{H}_g$ into that in 