On Multicommutators for Simple Algebraic Groups

There are several examples of groups for which any pair of commutators can be written such that both of them have a common entry, and one can look for a similar property for n-tuples of commutators. Here we answer, for simple algebraic groups over any field, the weaker question, under which condition the set of n-tuples of commutators with one common entry is Zariski dense in the set of all n-tuples of commutators. Surprisingly, there is a uniform bound on n in terms of the so-called Coxeter number of G to answer the question positively. An analogue result is proved for Lie algebras of simple and simply connected algebraic groups.