Comment by Gro-Tsen on Hilbert's Satz 90 for real simply-connected groups?

I'm not sure whether I misunderstood your question, but ⓐ to be clear, $\mathbb{R}$ does not satisfy the condition of having cohomological dimension $≤2$ (since its Galois group is finite cyclic nontrivial, and such have periodic cohomology), and ⓑ if $G$ is, for example, the simply-connected semisimple algebraic group $G_2$, then $H^1(k,G_2)$ classifies octonion algebras over $k$ (Serre, Galois Cohomology, III, appendix 2, 3.3), and for $k=\mathbb{R}$ there are two of them (the split octonions and usual octonions), so $H^1(\mathbb{R},G_2)$ is nontrivial.