Field-removing boosts for perpendicular electric and magnetic fields

For perpendicular electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$, the sign of $E^2-c^2{B}^2$ determines which field can be removed by a Lorentz boost. If positive, $\mathbf{w}=c^2\mathbf{E}\times \mathbf{B}/E^2$ gives ${\mathbf{B}}^{\prime }=0$; if negative, $\mathbf{w}=\mathbf{E}\times \mathbf{B}/B^2$ gives ${\mathbf{E}}^{\prime }=0$. Both results follow from ${\mathbf{E}}_{\perp }^{\prime }=\gamma (\mathbf{E}+\mathbf{w}\times \mathbf{B})$ and ${\mathbf{B}}_{\perp }^{\prime }=\gamma (\mathbf{B}-\mathbf{w}\times \mathbf{E}/c^2)$; the relevant strict inequality ensures $|\mathbf{w}|<c$. A nonzero null crossed field cannot be reduced to a single field by a finite Lorentz boost. A particle initially at rest in the laboratory has initial velocity $-\mathbf{w}$ in the chosen inertial frame, which must be retained when solving its trajectory.