= Pontryagin class
{c}
{title2=$p_k(E)=(-1)^kc_{2k}(E\otimes_{\mathbb R}\mathbb C)$}
For a real <vector bundle> $E$, its <Pontryagin classes> are
$$
p_k(E)=(-1)^k c_{2k}(E\otimes_{\mathbb R}\mathbb C)\in H^{4k}(X;\mathbb Z).
$$
They use the even-index <Chern classes> of the complexification. An integral qualification matters for <direct sums>: for the total <Pontryagin class>,
$$
2\bigl[p(E\oplus F)-p(E)p(F)\bigr]=0.
$$
Thus the usual multiplicative formula holds over the <rational numbers>, over the <real numbers>, or after discarding elements of order two. It need not hold without qualification in integral <cohomology>. The missing terms are products of odd <Chern classes> of complexified real <vector bundles>, which are annihilated by two.<Characteristic Classes by John Milnor and James D. Stasheff>[Chapter 15, Theorem 15.3]{ref}
For a metric <connection on a vector bundle> with <curvature matrix> $R$, a real representative is
$$
p_1(E)=-\frac{\operatorname{tr}(R\wedge R)}{8\pi^2}.
$$
The <Bianchi identity> makes this a <closed differential form>, and <Chern-Weil connection transgression> makes its <de Rham cohomology> class independent of the <connection on a vector bundle>. This normalization gives the degree-four <A-hat form> $-p_1/24$. Curvature representatives record real classes and therefore cannot detect the integral torsion qualification.
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