On relatively free subsets of Lie groups

In an arbitrary neighborhood $U$ of the identity $e$ of a connected Lie group there is a subset $S$ of cardinality $\mathfrak{c}$ and relatively free , i.e., the only nontrivial equations $x_1^{{\varepsilon _1}}x_2^{{\varepsilon _2}} \cdots x_n^{{\varepsilon _n}} = e,{\varepsilon _i} = \pm 1$, satisfied by substitution for distinct symbols among the ${x_i}$ distinct elements of $S$ are equations that are identities throughout $G$.