Curvature is C-infinity-linear in all three vector fields

Expanding the first slot by [F1]–[F3] gives $R(fX,Y)Z=f{\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-Y(f){\mathrm{\nabla }}_{X}Z-f{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-f{\mathrm{\nabla }}_{[X,Y]}Z+Y(f){\mathrm{\nabla }}_{X}Z\mathrm{.}$ The two derivative-of-$f$ terms cancel, leaving $fR(X,Y)Z$. The same calculation in the second slot gives $R(X,fY)Z=X(f){\mathrm{\nabla }}_{Y}Z+f{\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-f{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-f{\mathrm{\nabla }}_{[X,Y]}Z-X(f){\mathrm{\nabla }}_{Y}Z,$ and hence the second displayed identity.

For the third slot, two uses of the section Leibniz rule yield $R(X,Y)(fZ)=fR(X,Y)Z+(X(Yf)-Y(Xf)-[X,Y]f)Z\mathrm{.}$ The coefficient in parentheses is zero by the defining action of the Lie bracket on functions. Hence the third identity holds. Additivity and real homogeneity already follow from the connection and bracket laws, so these three identities prove separate $C^{\mathrm{\infty }}(M)$-linearity.