Following Warren Siegel's book on Field theory (pg. 223), one might derive the action of Lorentz generators $S_{ab}$ on an antisymmetric 2-tensor field strength $F_{cd}$ which arises for example in Maxwell field theory. $$(S_{ab} F)^{cd}=\delta_{[a}{}^{[c}F_{b]}{}^{d]}$$
Now, using a generic free field equation $$(S_a^b \partial_b+w \partial_a)F_{cd}=0$$ which is justified a few pages earlier, it is said that this equation reduces to
$$\frac{1}{2}\partial_{[a}F_{cd]}-\eta_{a[c}\partial^b F_{d]b}+(w-1)\partial_a F_{cd}=0$$
I have no clue on how to derive this equation. This should be a simple manipulation of tensor indices, but such a manipulation is not presented elsewhere in the book. I tried expanding out the antisymmetrizations in full but still I could not get the factor of $\frac{1}{2}$ in the first term or the last $(w-1)$ term. It is said that the irreducible pieces are separated out, by which I thought $$\partial_a F_{bc}=\partial_{[a}F_{bc]}+\text{symmetric part}+\text{trace}$$ is to be used, but I cannot see how to implement that.
It will be highly beneficial if someone can do the index manipulation in detail as the text is devoid of such examples and I cannot find an alternate textbook which discuss the topic in this manner and because I am new to this field.
