In fact, the free product of two finitely presented Lie algebras is also a finitely presented Lie algebra. Let consider the definition of dialgebras:
A diassociative algebra is a $K$-linear space, equipped with two $K$-line are maps $\dashv $ , $ \vdash$ : $D \times D \to D$ called respectively the left product and the right product such that the products $\dashv$ and $\vdash$ are associative and satisfy the following properties: $x \dashv (y \vdash z)= x \dashv (y \dashv z)$, $(x \dashv y) \vdash z= x \vdash (y \vdash z),$ $x \vdash (y \dashv z)= (x \vdash y) \dashv z$.
How can I show that whether the free product of two finitely presented free dialgebras is finitely presented or not?