The descent of the M2/M5-brane charges to the D\(p\)-brane charges and their twisted Bianchi identites \(d F_{p+2} =H_3 \wedge F_{p}\) is straightforward for \(p \in \{0,2,4\}\), as spelled out rigorously for instance in section 4.2 of

Varghese Mathai, Hisham Sati, *Some Relations between Twisted K-theory and E8 Gauge Theory*, JHEP0403:016,2004 (hep-th/0312033)

The origin of the D6 brane charge \(F_8\)is more subtle, as its M-theory lift involves the field of gravity: \(F_8 = \star_{10} F_2\). The correct Bianchi identity \(d F_8 = H_3 \wedge F_6\) is claimed on the left of (4.13) in the above article, but it does not actually follow from the 11d derivation given there, as far as I see.

What is a quick way to derive the Bianchi identity \(d F_8 = H_3 \wedge F_6\) in type IIA from 11d?