Freedom in conjugacy classes of simple algebraic groups and identities with constants

Let $G$ be a simple algebraic group defined over a field $k$, let $K/k$ be a field extension, and let $C_1,\dots,C_n$ be non-central conjugacy classes in $G(K)$. It is shown that if the transcendence degree tr.deg $K/k$ is sufficiently large, then almost always (except in the cases described) the elements $g_1\in C_1,\dots,g_n\in C_n$ in “general position” generate a subgroup of $G(K)$ isomorphic to the free-product $\langle g_1\rangle *\langle g_2\rangle *\dots *\langle g_n\rangle$ (modulo the center $Z(G(K))$. This result is deduced from another one, which deals with identities with constantsiiini the group $Z(G(K))$. Also, the case where $K=\overline Q$ is the algebraic closure of the field $Q$ of rational numbers is discussed.