Two short electrodynamics questions

1. Invariants in field strength

The first part of the question asked you to calculate the Lorentz scalars by contracting the field strength tensor (F) and it's dual (G): FF, GG and FG (index's omitted) and formed ±2(B²-E²/c²) and -4/c(E.B). The next part asked:

Are there any other invariants quadratic in the field strengths (but not depending on any higher derivatives of the potentials? Why or why not? [Hint: count parameters].


Attempt: I'm pretty sure the answer is no, as the invariants that can be formed from the field strength tensors are FF, GG, FG and GF (which gives nothing new). But I'm not sure how to show it formally. Is there are reason no other Lorentz scalars can be formed involving E & B?

2. Parity and time reversal of the Riemann-Silberstein vector

Question: What are the properties of the Riemann-Silberstein vector under parity and time reversal?

Equations: F(x,t) = E(x,t) + iB(x,t)

Attempt: Under parity reversal E -> -E and B -> B. Thus F -> -F*. This to me doesn't obviously make F a vector or pseudovector, have I made a mistake or is this another type of properties vectors can have or the extension of one of the other properties to complex numbers.

As for time reversal E -> E and B -> -B, so F -> F*. Again I'm not sure how to interpret this result.

Thanks for any help!
Any thoughts?

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