In Restricted Lorentz Transformation ${\mathrm{\Lambda }}^{-1}=\left[\begin{array}{cccc}\gamma & \beta \cdot \gamma & 0& 0\\ \beta \cdot \gamma & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

• ${\tilde{F}}_{0,1}$ = $F_{\alpha ,\beta }\cdot ({\mathrm{\Lambda }}^{-1}{)}_0^{\alpha }\cdot ({\mathrm{\Lambda }}^{-1}{)}_1^{\beta }$ = $({\mathrm{\Lambda }}^{-1}{)}_0^0\cdot ({\mathrm{\Lambda }}^{-1}{)}_1^0\cdot \cancel{F_{0,0}}+({\mathrm{\Lambda }}^{-1}{)}_0^0\cdot ({\mathrm{\Lambda }}^{-1}{)}_1^1\cdot F_{0,1}+({\mathrm{\Lambda }}^{-1}{)}_0^0\cdot \cancel{({\mathrm{\Lambda }}^{-1}{)}_1^2}\cdot F_{0,2}+({\mathrm{\Lambda }}^{-1}{)}_0^0\cdot \cancel{({\mathrm{\Lambda }}^{-1}{)}_1^3}\cdot F_{0,3}+({\mathrm{\Lambda }}^{-1}{)}_0^1\cdot ({\mathrm{\Lambda }}^{-1}{)}_1^0\cdot F_{1,0}+\dots$ → Simplify → = $({\mathrm{\Lambda }}^{-1}{)}_0^0\cdot ({\mathrm{\Lambda }}^{-1}{)}_1^1\cdot F_{0,1}+({\mathrm{\Lambda }}^{-1}{)}_0^1\cdot ({\mathrm{\Lambda }}^{-1}{)}_1^0\cdot F_{1,0}$ = $[\gamma \cdot \gamma \cdot \frac{E_x}{c}]+[(\gamma \cdot \frac{v}{c})\cdot (\gamma \cdot \frac{v}{c})\cdot (-\frac{E_x}{c})]$ = $\frac{E_x}{c}$ → $$\tilde{E}x = \tilde{F}\cdot c=E_x$$
• ${\tilde{F}}_{0,2}$ = #TODO
• $\dots$

The final result is: $$\begin{cases} \tilde{E}_x=E_x\ \tilde{E}_y = \gamma\cdot(E_y-v\cdot B_z)\ \tilde{E}_z = \gamma\cdot(E_z+v\cdot B_z)\end{cases} & \begin{cases} \tilde{B}_x=B_x\ \tilde{B}_y = \gamma\cdot(B_y+ \frac{v}{c^2}\cdot E_z)\ \tilde{B}_z = \gamma\cdot (B_z-\frac{v}{c^2}\cdot B_z)\end{cases}$$
Notice Electric Field $E$ & Magnetic Field $B$ are interconnect: 1 can $\mathrm{\exists }$ alone only IF IRF @rest!