Undecidability of the First Order Theories of Free Noncommutative Lie Algebras

Let R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article 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. Furthermore, if R has characteristic zero then we prove that the elementary theory of L in the standard ring language is undecidable. To do so we show that the arithmetic is 0-interpretable in L. This implies that the theory of has the independence property. These results answer some old questions on model theory of free Lie algebras.