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For the Bockstein associated with $0\to \mathbb{Z}/2\to \mathbb{Z}/4\to \mathbb{Z}/2\to 0$,For a real line bundle $L$, one first has $\beta (w_1(L))=w_1(L{)}^2$. The general identity follows from the splitting principle for real vector bundles, the Bockstein derivation rule, and the elementary-symmetric-polynomial identity obtained by separating whether an index lies in a chosen $j$-element subset.

Let $E$ have rank $r\ge 1$, let $p:\mathbb{P}(E)\to X$ be its projectivization of a real vector bundle, and let $\lambda \subset p^{\ast }E$ be its tautological bundle. Put $u=w_1(\lambda )\in H^1(\mathbb{P}(E);{\mathbb{F}}_2)$. The Projective bundle definition of Stiefel–Whitney classes starts from the isomorphismIn particular, $p^{\ast }$ is injective. Expand $u^r$ uniquely in this basis. The coefficients define the Stiefel–Whitney classes through the relationSet $w_0(E)=1$ and $w_i(E)=0$ for $i>r$; for a rank-zero vector bundle set $w(E)=1$. Uniqueness also proves naturality of the Stiefel–Whitney classes under a pullback vector bundle.

To prove the Whitney product formula for Stiefel–Whitney classes, apply the splitting principle for real vector bundles simultaneously to $E$ and $E^{\prime }$. Its injective pullback on mod-two cohomology follows by iterating the Projective bundle definition of Stiefel–Whitney classes. It suffices to compute for a split vector bundle $E={⨁}_{i=1}^r{L}_i$. Put $x_i=w_1(L_i)$.

On $\mathbb{P}(E)$, the coordinate projections of the tautological inclusion $\lambda \hookrightarrow p^{\ast }E$ give sections of the real line bundles ${\lambda }^{\ast }\otimes p^{\ast }{L}_i$ with no common zero. The mod-two Euler class of a real line bundle is its first Stiefel–Whitney class, and the First Stiefel–Whitney class of a tensor product of real line bundles gives $u+p^{\ast }{x}_i$. Multiplication of Euler classes under a direct sum, and vanishing of the Euler class of a vector bundle with a nowhere-zero section, therefore giveComparing with the unique monic relation defining the Stiefel–Whitney classes shows that $w_j(E)$ is the $j$th elementary symmetric polynomial in the $x_i$. Thus $w(E)={\prod }_i(1+x_i)$. A direct sum joins the two lists, so injectivity of the splitting pullback gives

For the immersion obstruction, use the notation $g=(f,f)$: the printed $f\times f$ here means the map from one copy of Real projective space into the product. Write $x=w_1({\gamma }_{\mathbb{R}})$ and let $z$ be the mod-two reduction of $c_1({\gamma }_{\mathbb{C}}^{\ast })$. The mod-two cohomology ring of real projective space and the cohomology ring of complex projective space giveNontriviality in degree two forces $f^{\ast }z=x^2$. Necessarily $k\ge 1$, since the degree-two cohomology of ${\mathbb{CP}}^0$ vanishes.

The standard tangent bundle identities for Real projective space and Complex projective space areFor the underlying real vector bundle of a complex line bundle, its total Stiefel–Whitney class is $1+c_1\mathrm{mod}2$. HenceIf $g$ is homotopic to an immersion $j$, its normal bundle $\nu$ has rank $4k-(2k+1)=2k-1$, andThe Whitney product formula for Stiefel–Whitney classes and homotopy invariance of cohomology yieldThe division is legitimate in the truncated cohomology ring, since $1+x$ is a unit. The degree-$2k$ component isIt must vanish because $2k>\mathrm{rank}\nu$, while $x^{2k}\ne 0$. Therefore $k$ must be odd.

For variables $x_1,\dots ,x_m$, the elementary symmetric polynomial and power-sum symmetric polynomial arePut $E(t)={\prod }_i(1+x_{i}t)={\sum }_{r\ge 0}{e}_r{t}^r$. Its logarithmic derivative isComparing the coefficient of $t^{n-1}$ in $E^{\prime }(t)=E(t)E^{\prime }(t)/E(t)$ gives the Newton identities

The given Bockstein homomorphism is the first Steenrod square. A real line bundle is pulled back from the universal line bundle over ${\mathbb{RP}}^{\infty }$, where the degree-one generator $x$ satisfies $\beta (x)=x^2$. By naturality,

Apply the splitting principle for real vector bundles, writing the pulled-back bundle as $L_1\oplus \cdots \oplus L_r$ and $x_i=w_1(L_i)$. The pullback in cohomology is injective, $w_j(E)=e_j(x_1,\dots ,x_r)$, and the Bockstein derivation rule givesIn $e_1{e}_j$, the terms for which the index from $e_1$ already lies in the $j$-element subset give this sum, while each square-free monomial of degree $j+1$ occurs $j+1$ times. Over ${\mathbb{F}}_2$ this yields

Let $p:\mathbb{P}(E)\to X$ be the projectivization of a real vector bundle, let $L_E$ be its tautological bundle, and putThe mod-two projective bundle formula says that $H^{\ast }(\mathbb{P}(E);{\mathbb{F}}_2)$ is a free $H^{\ast }(X;{\mathbb{F}}_2)$-module on $1,u,\dots ,u^{d-1}$. The Projective bundle definition of Stiefel–Whitney classes is the unique relation

Apply the splitting principle for real vector bundles. After an injective pullback, writeIf $x_i=w_1(L_i)$, the projective-bundle relation factors assoThe line summands of $E\oplus E^{\prime }$ are the union of the two lists, henceComparing the degree-$k$ components proves the Whitney product formula for Stiefel–Whitney classesInjectivity of the splitting pullback returns the identity to $X$.

For real line bundles, the transition functions take values in $O(1)=\{\pm 1\}$. Tensor product multiplies these signs, while the identification $\{\pm 1\}\cong \mathbb{Z}/2$ turns multiplication into addition. The corresponding degree-one characteristic classes therefore satisfy the First Stiefel–Whitney class of a tensor product of real line bundles formulaEquivalently, this follows from the classification of real line bundles by $H^1(X;{\mathbb{F}}_2)$.

Now take $M={\mathbb{RP}}^n$ and $E=k{\gamma }_{\mathbb{R}}^{1,n+1}$. Sincea line in $E_x$ is the fixed line ${\gamma }_x$ tensored with a line in ${\mathbb{R}}^k$. Thus the projectivization of copies of the real tautological line bundle isLet $x$ and $v$ be the degree-one generators pulled back from the first and second factors. The mod-two cohomology ring of real projective space and the Künneth theorem giveThe tautological line $L_E$ is the tensor product of the two tautological lines, so $w_1(L_E)=x+v$. In the alternative generator $u=w_1(L_E)$, the same ring is

The stable tangent-bundle identitygivesFor the vertical part of the tangent bundle of a projectivized real vector bundle,Each of the $k$ line summands on the right has first Stiefel–Whitney classThe Whitney product formula for Stiefel–Whitney classes therefore yieldsthe Total Stiefel–Whitney class of the projectivization of copies of the tautological line.

For real vector bundles $E$ and $F$,After applying the splitting principle for real vector bundles, both sides are the product of $1+w_1(L)$ over all line summands.