+ ...

The "reduced case" Let {1 , ••• {n be a basis of simple roots of gr;. It clearly defines on {)~ an equivalence relation (for any A, f.1 E {)~: A - f.1 if C, = f,J, gr; being canonically identified to the quotient space. Choose in each equivalence class C (i = 1, ... , 11) a representative element ai such that ai E gr {ai, i = 1, ... , n} is easily seen to be a basis of simple roots of an irreducible finite root system grr. isomorphic to gr; which is a subroot system of gr. grf is called the gradient root system.