Twisted tensor products related to the cohomology of the classifying spaces of loop groups

Let $G$ be a compact, simply connected, simple Lie group. We show that each twisted tensor product associated with the cohomology of $G$, in the sense of Brown, due to Kono, Mimura, Sambe and Shimada becomes that possessing a differential graded algebra structure in the sense of Hess. We thus obtain an economical injective resolution to compute, as an algebra, the cotorsion product which is the $E_2$-term of the cobar type Eilenberg-Moore spectral sequence converging to the cohomology of classifying space of the loop group $LG$. As an application, the cohomology $H^*(BLSpin(10); {\mathbb{Z}}/2)$ is explicitly determined as an $H^*(BSpin(10); {\mathbb{Z}}/2)$-module with the aid of the Hochschild spectral sequence and the TV-model for $BSpin(10)$.