The quaternionic Maxwell equations

Question

For the electromagnetic fields $(E,B)$, I propose to use the quaternionic numbers $i,j,k$:

$$E=E_w+i E_x +j E_y +k E_z$$

$$B=B_w+i B_x+j B_y +k B_z$$

$$q=w+ix+jy+kz$$

$$\frac{\partial}{\partial q}=\frac{\partial}{\partial w}-i\frac{\partial}{\partial x}-j\frac{\partial}{\partial y}-k\frac{\partial}{\partial z}$$

$$\frac{\partial}{\partial \bar q}=\frac{\partial}{\partial w}+i\frac{\partial}{\partial x}+j\frac{\partial}{\partial y}+k\frac{\partial}{\partial z}$$

The quaternionic Maxwell equations are:

$$\frac{\partial E}{\partial t}=\frac{\partial B}{\partial q}$$

$$div(E)=0$$

$$\frac{\partial B}{\partial t}=-\frac{\partial E}{\partial \bar q}$$

$$div(B)=0$$

Can we recover the classical Maxwell equations?

Comments

Why don't you simply check yourself? It appears rather straightforward to do so.

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It is not my problem.