Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup.
Given a $G$-torsor $E$ on $X$ in the etale topology, I would like to understand why it has a reduction to $B$, in the simple case above (usually one finds some generalizations).
I already understood that on some open dense subset the $G$-torsor trivializes, and hence has a $B$-reduction. The next step is to consider $B\backslash E \to X$, which is a fiber bundle in the etale topology, with fiber $B \backslash G$. If I understand correctly, it is clear what is $B \backslash E$ as an algebraic space - since $E$ is etale-locally trivial and algebraic spaces glue in the etale topology. I also read somewhere that in fact $B \backslash E$ is a scheme, but I don't know the reason, and it says it is not important for the argument.
Now, I would like to take a section of $B\backslash E \to X$ over some dense open subset, and continue it to the missing points, by valuative criterion of properness. The problem is that for algebraic spaces in the criterion there appears some finite field extension, and anyway I don't understand algebraic spaces to well. So the question is - could somebody explain how to technically proceed to show that one can extend the section to the missing points?
Thank you, Sasha