Let $R$ be a commutative ring with unit, and let $M$ be a CW complex. Let $\rho_R:H^*(M;\mathbb Z)\to H^*(M;R)$ denote the coefficient homomorphism induced by the unital ring map $\mathbb Z\to R$. Define the completed even cohomology group
degreewise as follows: for each $n\ge 0$, the homogeneous degree-$n$ part of $\prod_{i=1}^{r}Q(x_i)$, with each $x_i$ assigned cohomological degree $2$, is written uniquely as a polynomial in the elementary symmetric functions $e_1(x),\dots,e_r(x)$, and $K_Q(E)_n\in H^{2n}(M;R)$ is obtained by substituting $\rho_R(c_1(E)),\dots,\rho_R(c_r(E))$. Equivalently, this is the symmetric formal Chern-root expression
degreewise as follows: writing $q:=\lfloor d/2\rfloor$, for each $n\ge 0$ the homogeneous degree-$n$ part of $\prod_{j=1}^{q}P(y_j)$, with each $y_j$ assigned cohomological degree $4$, is written uniquely as a polynomial in the elementary symmetric functions $e_1(y),\dots,e_q(y)$, and its degree-$4n$ component is obtained by substituting $\rho_R(p_1(V)),\dots,\rho_R(p_q(V))$; its components in degrees $4n+2$ are zero. Equivalently, this is the symmetric formal Pontryagin-root expression