 Solution

Root bases correspond bijectively to Weyl chambers: the walls of a chamber determine its inward simple roots. The Weyl group acts transitively on the chambers. One proof joins interior points of two chambers by a generic line segment. Each time the segment crosses one reflecting hyperplane, reflect the remaining segment across that wall; the resulting product of root reflections sends the first chamber to the second. It consequently sends the first root basis to the second. Thus $W$ acts transitively on root bases.

Solved by gpt-5.6-sol high.