For $L/K$ an extension of char-$0$ nonarchimedean local fields, it is a classical theorem that for a torus T, if it is the product of the tori of the form $\text{Res}_{L/K}(G_{m})$, then $H^{1}(K,T)=1$.
For the norm one torus $\text{Res}^{1}_{L/K}(G_{m})$ in $\text{Res}_{L/K}(G_{m})$ defined via $$1\to \text{Res}^{1}_{L/K}(G_{m})\to \text{Res}_{L/K}(G_{m})\to G_{m}\to 1$$, it has nontrivial first Galois cohomology $$H^{1}(K,\text{Res}^{1}_{L/K}(G_{m}))\cong K^{*}/N_{L/K}L^{*}$$, and this follows from the long exact sequence for cohomology.
There is a result that any $L$-split $K$-torus $T$ can be written as $$T=G_{m}^{p}\times \text{Res}_{L/K}(G_{m})^{q}\times \text{Res}^{1}_{L/K}(G_{m})^{r}$$, for some uniquely determined positive integers $p,q,r$, so for $L$-split $K$-torus $T$, we have $H^{1}(K,T)=1$ if and only if $T$ is a product of tori of the form $\text{Res}_{L/K}(G_{m})$.
Now for $G,H$ two algebraic groups over $K$, we have the result that the $G(K)$-orbits in $(H\backslash G)(K)$ are parametrized by $\text{Ker}(H^{1}(K,H) \to H^{1}(K,G))$. Can we give some general criterions for this map being injective?