Differential second Bianchi identity

Expanding the alternating definition of $d^{\mathrm{\nabla }}R$ gives three output-derivative terms and the bracket terms $-R([X,Y],Z)+R([X,Z],Y)-R([Y,Z],X)$. Replace every bracket by $[A,B]={\mathrm{\nabla }}_{A}B-{\mathrm{\nabla }}_{B}A$ using [F4], and use skewness of the two-form $R$. The six resulting argument-derivative terms are exactly those subtracted in the tensor covariant derivatives, so $d^{\mathrm{\nabla }}R(X,Y,Z)=({\mathrm{\nabla }}_{X}R)(Y,Z)+({\mathrm{\nabla }}_{Y}R)(Z,X)+({\mathrm{\nabla }}_{Z}R)(X,Y)$.

Specialize [F1] to the tangent bundle to make the left side of step 1.1 zero, proving the first displayed identity. Since the Levi–Civita connection preserves $g$, lowering the output in [F2] commutes with covariant differentiation by [F3]; evaluating the resulting contracted identity on $U,V$ gives the second displayed formula.