I'm working on a paper from Steinberg (1974), "On a theorem of Pittie". The paper is mainly about roots and reflection groups. I'm having trouble understanding the proof of lemma 2.5(a) which says the following:
Let W be a Weyl group and let W' be an any reflection subgroup of W. R, R+, R' and R+' are now the corresponding (positive) root systems belonging to resp. W and W'. Let W" be the subset of W, which keeps R'+ positive. Thus W"={ w $\in$ W | w (R'+) $\subset$ R+ }.
The statement is: W" is a system of representatives for W/W'.
Proof by Steinberg:
Fix w $\in$ W. Then $w^{-1}$R+ $\cap$ R' and R'+ are both positive systems for R, hence $(*)$ they are congruent under a unique x $\in$ W'. Then u = w$x^{-1} \in$ W" and w = ux $\in$ W" $*$ W'. Conversely, if w has this form, we may work backwards to conclude that x satisfies $(*)$, hence is uniquely determined. This proves the statement.
I don't understand how this proof works. For starters, I don't think that $w^{-1}$R+ $\cap$ R' is a positive system. I understand that the different systems can be changed into one another when w works on it, but after that I can't follow the proof.
It would be very thankful if someone could clear this up! :)