= Solution
Contractions of <Lorentz tensors> have no free indices and are <Lorentz scalars>. The antisymmetry and the components in part (a) give
$$
\boxed{F_{\mu\nu}F^{\mu\nu}=2\left(|\boldsymbol B|^2-\frac{|\boldsymbol E|^2}{c^2}\right),\qquad
F_{\mu\nu}\widetilde F^{\mu\nu}=-\frac4c\boldsymbol E\cdot\boldsymbol B.}
$$
Thus $\boldsymbol E\cdot\boldsymbol B$ and $|\boldsymbol B|^2-|\boldsymbol E|^2/c^2$ are the two <electromagnetic field invariants> under proper <Lorentz transformations>. The second contraction uses the orientation-dependent epsilon tensor: strictly, $\boldsymbol E\cdot\boldsymbol B$ is a <pseudoscalar>, changing sign under an improper transformation such as spatial parity. Under the boosts in this question it is invariant.
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