I want to calculate $ A_{fgh} e^f \wedge e^g \wedge e^h$ where $A_{fgh}$ is some complicated term depending on 3 of the sub-indices. I am using the convention that 1 upper and 1 lower index gets summed over. I have the identities $\epsilon_{b}\mathstrut^{fgh} \epsilon^{b}\mathstrut_{fgh}=4!$ and $\star e^b = \frac{1}{3!}\epsilon^{b}\mathstrut_{fgh}e^f\wedge e^g \wedge e^h$. I want to know if it is valid to do the following.
$$ A_{fgh} \frac{1}{4!} \epsilon_{b}\mathstrut^{fgh} \epsilon^{b}\mathstrut_{fgh} e^f \wedge e^g \wedge e^h = \frac{1}{4} A_{fgh}\epsilon_{b}\mathstrut^{fgh}\star e^b.$$
What confuses me is using indices that are summed over. Is everything okay? Thanks in advance.
Explicity,
$$A^{a}_{fgh} := \epsilon^{ij}\mathstrut_{gh}F_{af}F_{ij} - \epsilon^{ij}\mathstrut_{ah}F_{fg}F_{ij}$$
where $F$ can be assumed to be antisymmetric if necessary and $a$ is independent. I essentially want to be able to write $A^{a}_{fgh,b}\star e^b.$
You can use an index twice, but not four times. It should be$$A_{fgh}e^f\wedge e^g\wedge e^h=A_{fgh}\tfrac{1}{4!}e_b^{\;ijk}e^b_{\;ijk}e^f\wedge e^g\wedge e^h.$$What may be more useful for you is to average resullts such as$$A_{fgh}e^f\wedge e^g\wedge e^h=A_{gfh}e^g\wedge e^f\wedge e^h=-A_{gfh}e^f\wedge e^g\wedge e^h$$to get$$A_{fgh}e^f\wedge e^g\wedge e^h=A_{[fgh]}e^f\wedge e^g\wedge e^h.$$