Pontryagin classes by complexification

Assume AC. Let $E\to B$ be a real vector bundle of rank $n$ over a path-connected CW base (or a CW-type base), with complexification $E_{\mathbb{C}}$ and Chern classes $c_i(E_{\mathbb{C}})\in H^{2i}(B;\mathbb{Z})$ in the sense of Chern classes from the projective-bundle relation. The Pontryagin classes of $E$ are defined by $p_i(E):=(-1{)}^i{c}_{2i}(E_{\mathbb{C}})\in H^{4i}(B;\mathbb{Z})(i\ge 0),$ completed by the conventions $p_0(E):=1$ and $p_i(E):=0$ whenever $2i>n={\mathrm{rank}}_{\mathbb{R}}E$; the total Pontryagin class is the finite sum $p(E):={\sum }_{i\ge 0}{p}_i(E)$.

The definition is well defined and requires no orientation of $E$: the complexification $E_{\mathbb{C}}$ is determined by $E$ up to canonical isomorphism, so its even Chern classes are determined, and the sign $(-1{)}^i$ is a fixed normalization (it makes $p_i$ of a line vanish and makes the top class $p_n$ of an oriented rank-$2n$ bundle equal to $e(E{)}^2$ in the next items). When $E$ is numerable and $B$ is a path-connected paracompact Hausdorff CW complex, the odd Chern classes of $E_{\mathbb{C}}$ are two-torsion by Odd Chern classes of a complexified real bundle are two-torsion and enter no Pontryagin class; conjugation invariance of the even classes, $c_{2i}(\overline{E_{\mathbb{C}}})=c_{2i}(E_{\mathbb{C}})$, is what makes the construction orientation-free.