Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 3 / b

Regard an $E^{\vee }$-valued $r$-form $\lambda$ as a row vector and an $\mathrm{End}(E)$-valued $r$-form $\mu$ as a matrix. The dual connection and endomorphism connection areFor the curvature two-form,because substituting $F=dA+A\wedge A$ makes all terms cancel. This is the Bianchi identity.

For an endomorphism-valued $r$-form $\mu$ and an $E$-valued form $\sigma$, direct expansion gives the compatible Leibniz ruleThe cancellation of the two middle $A$-terms proves the identity.