  Let $R$ be a commutative integral unital domain and $L$ a free
non-commutative Lie algebra over $R$. In this paper we show that the ring $R$
and its action on $L$ are 0-interpretable in $L$, viewed as a ring with the
standard ring language $+, \cdot,0$. Furthermore, if $R$ has characteristic
zero then we prove that the elementary theory $Th(L)$ of $L$ in the standard
ring language is undecidable. To do so we show that the arithmetic ${\bf N} =
\langle{\bf N}, +,\cdot,0 \rangle$ is 0-interpretable in $L$. This implies that
the theory of $Th(L)$ has the independence property. These results answer some
old questions on model theory of free Lie algebras.
