Let $A,B,C,D$ be algebraic systems and $A$ and $B$ be elementary equivalent as well as $C$ and $D$. Are free products of $A,C$ and $B,D$ elementary equivalent if
- $A,B,C,D$ are groups, or
- $A,B,C,D$ are Lie algebras?
If one of this problem (or both) has already been solved, it would be nice to get a reference to the corresponding paper(s).
