Complexification is conjugation invariant

Complexification is constructed by viewing the real transition matrices of $E$ as complex matrices; the same cocycle and numeration define a complex vector bundle by [F1]. Its fibers identify with $E_b{\otimes }_{\mathbb{R}}\mathbb{C}$. The map $\phi :E_{\mathbb{C}}\to \overline{E_{\mathbb{C}}}$, $v\otimes z\mapsto v\otimes \bar{z}$, is complex-linear: $\phi (i(v\otimes z))=\phi (v\otimes iz)=v\otimes \overline{iz}=-i(v\otimes \bar{z})$ while in the conjugate structure $i\cdot \phi (v\otimes z)$ is by definition $-i(v\otimes \bar{z})$; real balancing makes this tensor formula well defined. In every real bundle chart it is ordinary coordinate conjugation, which commutes with real transition matrices. It and its inverse are continuous by that same local formula, so it is an isomorphism of complex bundles. This is assertion 1.

For a complex line $L$, conjugating transition functions inverts the first Chern class: $c_1(\bar{L})=-c_1(L)$, which identifies the conjugate of each root with its negative.

First suppose $V$ has rank $n\ge 1$. Let $q:F\to B$ be its flag splitting from [F3], and choose a homotopy equivalence $h:W\to F$ from a path-connected CW complex with a vertex. Write $r=qh$ and $K_j=h^{\ast }{L}_j$, so $r^{\ast }V={⨁}_j{K}_j$ and $r^{\ast }\bar{V}={⨁}_j\overline{K_j}$ by the transition formulas of [F1]. These are bundles over the actual CW base $W$, where [F3] gives Whitney multiplication. Put $s_j=c_1(K_j)$. Naturality and step 1.2 give $r^{\ast }c(V)={\prod }_j(1+s_j)$ and $r^{\ast }c(\bar{V})={\prod }_j(1-s_j)$. Every degree-$i$ elementary symmetric monomial has exactly $i$ factors, so $r^{\ast }{c}_i(\bar{V})=(-1{)}^i{r}^{\ast }{c}_i(V)$. No root transformation is attributed to a metric theorem, and no CW-only Whitney interface is applied on the merely CW-type space $F$.

The map $r^{\ast }=h^{\ast }{q}^{\ast }$ is injective, because $q^{\ast }$ is injective by [F3] and $h^{\ast }$ is an isomorphism by [F4]. Thus the equality in step 2.1 descends to $c_i(\bar{V})=(-1{)}^i{c}_i(V)$. If $V$ has rank zero, its conjugate is again zero and [F3] gives $c_0=1$ and all higher classes zero, so the same conclusion holds without a flag construction.

Applying assertion 2 to $V=E_{\mathbb{C}}$ and using the canonical isomorphism of step 1.1 gives $c_i(E_{\mathbb{C}})=c_i(\overline{E_{\mathbb{C}}})=(-1{)}^i{c}_i(E_{\mathbb{C}})$, so the odd Chern classes of a complexified real bundle are two-torsion; this consistency is used in the next items.

Boundary cases. Rank zero was settled in step 3.1. For $i=0$ both sides are $1$; for a rank-one $V$ the statement is step 1.2. The empty base is excluded by the path-connected hypothesis, and the coefficient ring $\mathbb{Z}$ is nonzero. The metric is used only to identify $\bar{L}$ with $L^{\ast }$ inside [F2]; no orientation or choice of frames enters. AC is used only through [A1] in the splitting and metric suppliers.