  In this paper we study the Diophantine problem in Chevalley groups $G_\pi
(\Phi,R)$, where $\Phi$ is an indecomposable root system of rank $> 1$, $R$ is
an arbitrary commutative ring with $1$. We establish a variant of double
centralizer theorem for elementary unipotents $x_\alpha(1)$. This theorem is
valid for arbitrary commutative rings with $1$. The result is principle to show
that any one-parametric subgroup $X_\alpha$, $\alpha \in \Phi$, is Diophantine
in $G$. Then we prove that the Diophantine problem in $G_\pi (\Phi,R)$ is
polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine
problem in $R$. This fact gives rise to a number of model-theoretic corollaries
for specific types of rings.
