Let $M$ be a paracompact smooth manifold, and let $E\to M$ and $F\to M$ be smooth real vector bundles of finite rank. For each smooth real vector bundle $V\to M$ of rank $r$, let $p_k(V)\in H_{\mathrm{dR}}^{4k}(M)$ denote its $k$-th Pontryagin class, with $p_0(V)=1$ and $p_k(V)=0$ for $2k>r$. Define its total Pontryagin class by

\begin{align*} p(V):=\sum_{k\ge 0}p_k(V)\in H_{\mathrm{dR}}^{4*}(M), \end{align*}

where $H_{\mathrm{dR}}^{4*}(M):=\bigoplus_{k\ge 0}H_{\mathrm{dR}}^{4k}(M)$ and multiplication is the cup product in the graded real de Rham [cohomology ring](/theorems/2271). Then

\begin{align*} p(E\oplus F)=p(E)p(F) \end{align*}

in $H_{\mathrm{dR}}^{4*}(M)$. Equivalently, for every integer $k\ge 0$,

\begin{align*} p_k(E\oplus F)=\sum_{i+j=k}p_i(E)p_j(F) \end{align*}

in $H_{\mathrm{dR}}^{4k}(M)$.