Bott vanishing for real Pontryagin monomials of a foliation

By [F4] choose a metric on $TM$, the orthogonal projection $\mathrm{\Pi }:TM\to TF$, and a connection ${\mathrm{\nabla }}^0$ on $\nu$. Define ${\mathrm{\nabla }}_{X}s={\mathrm{\nabla }}_{\mathrm{\Pi }X}^{B}s+{\mathrm{\nabla }}_{X-\mathrm{\Pi }X}^{0}s$. Its direction-linearity and Leibniz rule follow from those of the two summands, since $(\mathrm{\Pi }X)(f)+(X-\mathrm{\Pi }X)(f)=X(f)$; it extends the Bott partial connection. Now [F1] puts all its curvature entries in $I$ with $I^{q+1}=0$.

Evaluating the invariant polynomial $\phi$ of degree $k$ on the curvature gives a sum of products of $k$ matrix entries ${\mathrm{\Omega }}^{a_i}{}_{b_i}$, possibly with constant coefficients and traces; each such product is an element of $I^k$, so $\phi (R^{\mathrm{\nabla }})\in I^k$.

If $k>q$ then $I^k\subseteq I^{q+1}=0$, so the Chern-Weil form $\phi (R^{\mathrm{\nabla }})$ is identically zero and represents the zero class; by [F2] the Chern-Weil class is independent of the connection and by [F3] it represents the real characteristic class of $\nu$, so the real Pontryagin monomial $\phi (\nu )$ vanishes in $H_{\mathrm{d}\mathrm{R}}^{2k}(M;\mathbb{R})$; no integral refinement is claimed with AC used for [F4] and the topological-to-real comparison [F3].