TITLE: A connected reductive algebraic group over a separably closed field is a rational variety
QUESTION [6 upvotes]: I need either a proof or a reference in modern (scheme-theoretic) language. According to Sansuc, this result can be gleaned from Borel's book on linear algebraic groups, but the old-style algebraic geometry language makes my head spin. Isn't there a modern proof of the above statement somewhere, for goodness sake?

REPLY [3 votes]: How's about these notes by Gille?
And these notes by Colliot-Thelene.
And these notes by Brian Conrad (Prop 7.2.3)

REPLY [2 votes]: This is 16.56 + 21.56 of J.S. Milne, Algebraic Groups: The theory of algebraic group schemes over a field, Cambridge U. P., 2017 (available September)