The L-polynomials are well defined and form a multiplicative, natural and stable sequence

For fixed $k$, take $m,n\ge \max (1,k)$ and let $x_1,\dots ,x_m,y_1,\dots ,y_n$ be indeterminates; put $A={\prod }_{i\le m}Q(x_i)$ and $B={\prod }_{j\le n}Q(y_j)$ in $\mathbb{Q}\mathrm{⟦}x,y\mathrm{⟧}$. Then $({\prod }_{i}Q(x_i))({\prod }_{j}Q(y_j))=AB$, so its homogeneous component of weight $4k$ is the finite sum ${\sum }_{i+j=k}([\text{weight }4i]A)([\text{weight }4j]B)$; the elementary symmetric function of the combined squared variables is $e_r(x_1^2,\dots ,x_m^2,y_1^2,\dots ,y_n^2)={\sum }_{a+b=r}{e}_a(x^2)e_b(y^2)$ by [F3]. Applying the defining identity of [F1] to the combined family of roots and to each block separately, we obtain the universal identity $L_k({\sum }_{a+b=1}{e}_a(x^2)e_b(y^2),\dots ,{\sum }_{a+b=k}{e}_a(x^2)e_b(y^2))={\sum }_{i+j=k}{L}_i(e(x^2))L_j(e(y^2))$ in the symmetric polynomial ring of the two blocks.

Well-definedness and trivial values: $L_k$ is a well-defined homogeneous weight-$4k$ polynomial in $p_1,\dots ,p_k$, independent of the number $N$ of formal roots, by the existence and uniqueness argument of the definition of [F1] together with the injectivity in [F2]; $L_0=1$ is the weight-$0$ component. Evaluating the defining identity at $x_1=0$ (so all $e_j(0)=0$ for $j\ge 1$) gives $L_k(0,\dots ,0)=[\text{weight }4k]Q(0)=0$ for $k\ge 1$; hence for a trivial bundle, whose positive Pontryagin classes vanish by [F4], the total class is $L({\epsilon }^r)=(1,0,0,\dots )$, the unit of the completed ring.

Rank two: let $E\to B$ be a numerable oriented real bundle of rank two over a path-connected paracompact Hausdorff CW base, with Euler class $e$. By [F7] $p_1(E)=e^2$, and $p_i(E)=0$ for $i\ge 2$ since $2i>\mathrm{rank}E$ by [F4]. Substituting into the defining identity of [F1] with $N=1$ gives $L_k(E)=L_k(e^2,0,\dots ,0)=[\text{weight }4k]Q(x)=q_{2k}{e}^{2k}$ after putting $x^2=e^2$, so $L(E)={\sum }_{k\ge 0}{q}_{2k}{e}^{2k}=Q(e)=e\mathrm{/}\tanh e$.

To extend the bundle identities to a path-connected paracompact Hausdorff CGWH base $B$ of CW type, use homotopy inverse maps $h:K\to B$, $g:B\to K$ from a CW model. The CW-type transport in Pontryagin numbers of a closed oriented manifold gives $p_i(E)=g^{\ast }{p}_i(h^{\ast }E)$, naturality between these bases, and stability. The Whitney identity for $h^{\ast }E,h^{\ast }F$ on $K$ therefore pulls back to $B$, and all polynomial L-identities do too. Euler and Chern naturality in [F9] give $e(E)=g^{\ast }e(h^{\ast }E)$ and the analogous Chern identity, so the rank-two and complex-line formulas also transport. For a disjoint union, each singular simplex lies in one component; cochains, their differentials and cup products are componentwise products, hence every class and identity assembles componentwise, without selecting representatives for a family of classes.

Multiplicativity: since the two blocks $e_a(x^2)$ and $e_b(y^2)$ are jointly algebraically independent by two applications of [F2], the identity of step 1.1 is equivalent, under the inverse substitution $P_a\mapsto e_a(x^2)$, $P_b^{\mathrm{\prime }}\mapsto e_b(y^2)$, to the polynomial identity $L_k(P_1^{\mathrm{\prime }\mathrm{\prime }},\dots ,P_k^{\mathrm{\prime }\mathrm{\prime }})={\sum }_{i+j=k}{L}_i(P)L_j(P^{\mathrm{\prime }})$ in $\mathbb{Q}[P_1,\dots ,P_k,P_1^{\mathrm{\prime }},\dots ,P_k^{\mathrm{\prime }}]$, where $P_r^{\mathrm{\prime }\mathrm{\prime }}:={\sum }_{a+b=r}{P}_a{P}_b^{\mathrm{\prime }}$. For bundles $E,F$ over $B$, [F5] gives $p_r(E\oplus F)={\sum }_{a+b=r}{p}_a(E)p_b(F)$ in $H^{4r}(B;\mathbb{Q})$, so substituting $P_a\mapsto p_a(E)$, $P_b^{\mathrm{\prime }}\mapsto p_b(F)$ into that polynomial identity yields $L_k(E\oplus F)={\sum }_{i+j=k}{L}_i(E)L_j(F)$ for every $k$; collecting degrees via the convolution product of [F6] gives $L(E\oplus F)=L(E)L(F)$ in ${\widehat{H}}^{4\ast }(B;\mathbb{Q})$.

Naturality: for a continuous $f:B^{\mathrm{\prime }}\to B$ and a bundle $E\to B$, [F4] gives $p_i(f^{\ast }E)=f^{\ast }{p}_i(E)$; since $f^{\ast }$ is a unital ring homomorphism on completed cohomology by [F6] and $L_k$ is a polynomial, $L_k(f^{\ast }E)=L_k(f^{\ast }{p}_1(E),\dots )=f^{\ast }{L}_k(E)$, and componentwise $L(f^{\ast }E)=f^{\ast }L(E)$.

Stability: [F4] gives $p_i(E\oplus {\epsilon }^r)=p_i(E)$ for every $i$, so $L_k(E\oplus {\epsilon }^r)=L_k(E)$ for every $k$ and $L(E\oplus {\epsilon }^r)=L(E)$; the trivial-bundle case is step 1.2.

Complex line: let $\mathrm{\ell }\to B$ be a complex line bundle with $c_1(\mathrm{\ell })=t$, regarded as an oriented real rank-two bundle through the complex orientation. By [F8] its Euler class is $e({\mathrm{\ell }}_{\mathbb{R}})=c_1(\mathrm{\ell })=t$, so step 1.3 gives $L({\mathrm{\ell }}_{\mathbb{R}})=Q(t)=t\mathrm{/}\tanh t$.

Whitney sums of rank-two bundles: iterating step 2.1 and using step 1.3, an oriented Whitney sum $E_1\oplus \cdots \oplus E_r$ of numerable oriented rank-two bundles has $L(E_1\oplus \cdots \oplus E_r)={\prod }_{i=1}^{r}L(E_i)={\prod }_{i=1}^{r}Q(e_i)$, with $e_i$ the Euler class of $E_i$.

Steps 1.2, 2.2 and 2.3 give well-definedness, naturality, stability and the trivial-bundle value; step 2.1 gives multiplicativity; steps 1.3, 2.4 and 3.1 give the rank-two, complex-line and Whitney-sum evaluations. All statements are identities between prescribed polynomials in characteristic classes over $\mathbb{Q}$; the empty Whitney sum is the trivial bundle, and the rank-zero and rank-two boundary cases are included by $L_0=1$ and by $p_i=0$ for $2i>\mathrm{rank}$, and AC is used only as declared through the Pontryagin-class construction and its multiplicativity supplier.