The notion of an excellent quadratic form has been introduced by M. Knebusch 10, 11]. It has an obvious analogue for a semisimple linear algebraic group G over an arbitrary eld k: the group G is excellent if, for any eld extension K of k, the anisotropic kernel of the K-group G k K can be deened by a k-group. The aim of this paper is to investigate excellence properties of special linear and or...