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The descent of the M2/M5-brane charges to the Dp-brane charges and their twisted Bianchi identites $\mathrm{d} F_{p+2} = H_3 \wedge F_p$ is straightforward for $p = 0,2,4$.

(This is 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)

The M-theory origin of the D6 brane charge $F_8$ is more subtle, as its M-theory lift involves the field of gravity. The derivation of the correct Bianchi identity $\mathrm{d} F_8 = H_3 \wedge H_6$ from 11d needs to use the SUGRA equations of motion.

What is a quick way to derive the Bianchi identity $\mathrm{d} F_8 = H_3 \wedge F_6$ in type IIA from 11d?

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