The L-genus is an oriented rational bordism ring homomorphism

Additivity and orientation: let $M,N$ be closed oriented $4k$-manifolds. The restriction of the tangent bundle of $M\bigsqcup N$ to $M$ is $TM$ and to $N$ is $TN$, so the polynomial $L_k(T(M\bigsqcup N))$ restricts to $L_k(TM)$ and $L_k(TN)$; with $[M\bigsqcup N]=[M]+[N]$ by [F4] this gives $L[M\bigsqcup N]=L[M]+L[N]$. For the opposite orientation, $T(-M)=TM$ (the tangent bundle does not see the orientation), while $[-M]=-[M]$ by [F4], so $L[-M]=-L[M]$.

Boundary vanishing: if $M=\mathrm{\partial }W$ with $W$ compact oriented and $\dim M=4k$, then, expanding by [F1], $L[M]={\sum }_{\mathrm{\mid }J\mathrm{\mid }=k}{c}_J{p}_J[M]={\sum }_{\mathrm{\mid }J\mathrm{\mid }=k}{c}_J\cdot 0=0$ because every Pontryagin number of a boundary vanishes by [F3].

Let $M^{4a},N^{4b}$ be closed oriented with the product orientation, and write ${\pi }_M,{\pi }_N$ for the projections. The tangent splitting [F5] and naturality and multiplicativity [F2] give $L(T(M\times N))={\pi }_M^{\ast }L(TM){\pi }_N^{\ast }L(TN)$. Its degree-$4(a+b)$ part is ${\sum }_{i+j=a+b}{L}_i(TM)\times L_j(TN)$. By [F10] only $i=a,j=b$ can survive. Evaluating this term on $[M]\times [N]$ using [F6] and the matching-degree identity [F7] gives $L[M\times N]=L[M]L[N]$. This uses the class identity from [F8], without its separate monomial-number expansion.

Multiplicativity with the zero convention: if $4\nmid (m+n)$, the product and at least one factor have value zero by [F1]. Otherwise suppose $m=\dim M$ is not divisible by four, write $m=4p+r$ with $1\le r\le 3$, and let $N$ have dimension $n=4k-m$ so that $M\times N$ has dimension $4k$. Then $L[M]=0$ by [F1]. In the degree-$4k$ part of $L(TM)L(TN)$ only the terms $L_i(TM)L_j(TN)$ with $i+j=k$ occur, and $L_i(TM)\in H^{4i}(M;\mathbb{Q})=0$ whenever $4i>m$, so $i\le p$; similarly $L_j(TN)=0$ whenever $4j>n=4(k-p)-r$, so $j\le k-p-1$ because $r\ge 1$; thus $i+j\le k-1<k$ and no term of total degree $4k$ survives. Hence $L[M\times N]=0=L[M]L[N]$, and the same argument with the roles of $M$ and $N$ interchanged covers $4\nmid \dim N$.

Descent: the value $L[M]$ is additive under disjoint union and vanishes on oriented boundaries by steps 1.1 and 1.2, so it is constant on oriented cobordism classes: for a cobordism $V$ from $M_0$ to $M_1$ with $\mathrm{\partial }V=-M_0\bigsqcup M_1$ by [F9], step 1.2 gives $0=L[\mathrm{\partial }V]=L[-M_0\bigsqcup M_1]=-L[M_0]+L[M_1]$ by step 1.1. Hence $L$ induces a well-defined additive map ${\mathrm{\Omega }}_n^{SO}\to \mathbb{Q}$ for each $n$ and, extended $\mathbb{Q}$-linearly, a functional on ${\mathrm{\Omega }}_{\ast }^{SO}\otimes \mathbb{Q}$ that assigns the value $0$ in degrees not divisible by four. It is unital: $L[\mathrm{p}\mathrm{t}]=1$ because the tangent bundle of a point is trivial and $L$ of a trivial bundle is $1$ by [F2].

By step 2.1 the functional is defined and additive on the rationalized oriented bordism groups, and by steps 1.3 and 1.4 it is multiplicative on products of closed oriented manifolds, with unit value $1$; since products of such classes generate ${\mathrm{\Omega }}_{\ast }^{SO}\otimes \mathbb{Q}$ biadditively, this makes $L$ a unital $\mathbb{Q}$-algebra homomorphism ${\mathrm{\Omega }}_{\ast }^{SO}\otimes \mathbb{Q}\to \mathbb{Q}$, assigning each closed oriented $4k$-manifold its L-genus and the value $0$ in dimensions not divisible by four. The empty disjoint union and the zero class satisfy both sides trivially.