Eisenstein cohomology for orthogonal groups and the special values of $L$-functions for ${\mathrm{G}\mathrm{L}}_1\times \mathrm{O}(2n)$

For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group $\mathrm{O}(2n+2)$ over a totally real number field $F\mathrm{.}$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands $L$-functions associated to the group ${\mathrm{G}\mathrm{L}}_1\times \mathrm{O}(2n)$ over $F$. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin - Selberg $L$-functions attached to a pair of Hilbert modular forms.