In #2016-413 the authors write in Equation 8 that for the Weil Pairing e_{mn}(P, Q)^n = e_m([n]P, [n]Q),
which is argued for by referring to [35, Lemma 16.2]. This citation is stated as

J.S. Milne. Arithmetic Geometry, chapter Abelian Varieties, pages 103–150. Springer New York, New York, NY, 1986.

Chasing this citation, I think it’s Cornell & Silverman’s Arithmetic Geometry (1986) with specifically chapter V by Milne, and so this seems to be the relevant chapter, with Lemma 16.2 as

Now for the question: the derivation from Lemma 16.2 to Equation 8 is for me not clear. Could anybody explain in a bit more elliptic-curve-specific language how one implies the other?

I’m not sure if you’re looking for a derivation from Lemma 16.2 to Equation 8 specifically or you’re looking for a proof of Equation 8.

In the second case, it can be proved using Proposition 8.1.e in Chapter 3 of Silverman’s The Arithmetic of Elliptic Curves (p. 94 in the second edition). It states that the Weil pairing is compatible:

Thanks! That is what I was looking for and I’ll dig in Silverman a bit. But I’ll leave this one open for anyone that wants to contribute a derivation of (e) from 16.2