The Waring problem for Lie groups and Chevalley groups

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given non-trivial word w. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed non-trivial word w and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)^2=G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over R or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.