 Solution

Choose, for example,The fields are perpendicular and satisfy $E_0^2=c^2{B}_0^2$, so both invariants vanish. Every nonzero constant background with vanishing invariants can be brought to this form by a spatial rotation, up to signs.

In this frame, and with a convention consistent with the relativistic Lorentz force, the mixed tensor isCalling the displayed dimensionless matrix $N$, direct multiplication givesConsequentlyThis is a tensor equation. Since it holds in the canonical frame, it holds in every Lorentz frame for any null electromagnetic field.