  The root systems appearing in the theory of Lie superalgebras and Nichols
algebras admit a large symmetry extending properly the one coming from the Weyl
group. Based on this observation we set up a general framework in which the
symmetry object is a groupoid. We prove that in our context the groupoid is
generated by reflections and Coxeter relations. This answers a question of
Serganova. Our weak version of the exchange condition allows us to prove
Matsumoto's theorem. Therefore the word problem is solved for the groupoid.
