Variables
$E^i =$ electric field
$B^i =$ magnetic field
$\rho =$ charge density
$j^i =$ current density
$\epsilon_0 =$ vacuum permittivity
$\mu_0 =$ vacuum permeability

Maxwell Equations
${E^i}_{,i} = \frac{1}{\epsilon_0} \rho$
${B^i}_{,i} = 0$
$\epsilon_0 {E^i}_{,t} - \frac{1}{\mu_0} {\epsilon^{ij}}_k {B^k}_{,j} = -j^i$
${B^i}_{,t} + {\epsilon^{ij}}_k {E^k}_{,j} = 0$

Let $q = \hat{q} exp(i \tilde{q}_\mu x^\mu)$ for $x^\mu \in \{ t,x,y,z \}$.
Therefore $q_{,\mu} = \tilde{q}_\mu q$.

$\tilde{E}_i E^i = \frac{1}{\epsilon_0} \rho$
$\tilde{B}_i B^i = 0$

$\epsilon_0 \tilde{E}_t E^i - \frac{1}{\mu_0} {\epsilon^{ij}}_k \tilde{B}_j B^k = -j^i$
$E^i = \frac{1}{\epsilon_0 \tilde{E}_t} (\frac{1}{\mu_0} {\epsilon^{ij}}_k \tilde{B}_j B^k - j^i)$

$\tilde{B}_t B^i + {\epsilon^{ij}}_k \tilde{E}_j E^k = 0$
$B^i = -\frac{1}{\tilde{B}_t} {\epsilon^{ij}}_k \tilde{E}_j E^k$
Substitute:
$B^i = -\frac{1}{\tilde{B}_t} {\epsilon^{ij}}_k \tilde{E}_j \frac{1}{\epsilon_0 \tilde{E}_t} (\frac{1}{\mu_0} {\epsilon^{kl}}_m \tilde{B}_l B^m - j^k)$
$B^i = -\tilde{E}_j \frac{1}{\epsilon_0 \tilde{E}_t \tilde{B}_t} ( \frac{1}{\mu_0} {\epsilon^{ij}}_k {\epsilon^k}_{lm} \tilde{B}^l B^m - {\epsilon^{ij}}_k j^k)$
$B^i = -\tilde{E}_j \frac{1}{\epsilon_0 \tilde{E}_t \tilde{B}_t} ( \frac{1}{\mu_0} ( \delta^i_l \delta^j_m - \delta^i_m \delta^j_l )\tilde{B}^l B^m - {\epsilon^{ij}}_k j^k)$
$ ( (\mu_0 \epsilon_0 \tilde{E}_t \tilde{B}_t - \tilde{E}_k \tilde{B}^k) \delta^i_j + \tilde{B}^i \tilde{E}_j ) B^j = \mu_0 {\epsilon^{ij}}_k \tilde{E}_j j^k$
Let ${A^i}_j = (\mu_0 \epsilon_0 \tilde{E}_t \tilde{B}_t - \tilde{E}_k \tilde{B}^k) \delta^i_j + \tilde{B}^i \tilde{E}_j$
${A^i}_j B^j = \mu_0 {\epsilon^{ij}}_k \tilde{E}_j j^k$
$B^l = \mu_0 {(A^{-1})^l}_i {\epsilon^{ij}}_k \tilde{E}_j j^k$

$E^i = \frac{1}{\epsilon_0 \tilde{E}_t} (\frac{1}{\mu_0} {\epsilon^{ij}}_k \tilde{B}_j B^k - j^i)$
Substitute $B^k = \mu_0 {(A^{-1})^k}_l {\epsilon^{lm}}_n \tilde{E}_m j^n$.
$E^i = \frac{1}{\epsilon_0 \tilde{E}_t} ({\epsilon^{ij}}_k \tilde{B}_j {(A^{-1})^k}_l {\epsilon^{lm}}_n \tilde{E}_m - \delta^i_n) j^i$