Corollary

— Let $V_0$ be a vector space over $K$ and let $W$ be a vector sub-N-space of $N \otimes_K V_0$. Suppose that $W$ is stable under the mappings $\sigma \otimes \mathrm{Id}_{V_0}$ for all $\sigma \in \Gamma$. Let $W_0$ be the set of $x \in V_0$ such that $1 \otimes x \in W$; then $W_0$ is the unique vector sub-K-space of $V_0$ such that $W = N \otimes_K W_0$.

It suffices to remark that the set of elements of the form $1 \otimes x \ (x \in V_0)$ is a K-structure on $N \otimes_K V_0$ for which we have $u_\sigma = \sigma \otimes \mathrm{Id}_{V_0}$ for $\sigma \in \Gamma$.