Let $X$ be a variety over a field $k$ of characteristic 0 (not necessarily with $k = \bar{k}$). Let $G_1, G_2$ be linear algebraic groups over $k$. My question is:
Can I write $H^1(X, G_1 \times G_2)$ in terms of $H^1(X, G_1)$ and $H^1(X, G_2)$ (or any other "simpler" things)?