In Higgins's paper Baer invariant and the Birkhoff-Witt theorem, the following definition is given: A Lie structure over the $R-$ module $M$ is a $T(M)$ bimodule $A$ together with a bilinear function $M\otimes M\to A$ taking $x \otimes y \to <x,y>$ satisfying $<x,x>=0$,

$<x,y>t(uv-vu)=(xy-yx)t(<u,v>) $ for all $x,y,u,v \in M$ and $t \in T(M)$ and

$(<x,y>z-z<x,y>)+(<y,z>x-x<y,z>)+(<z,x>y-y<z,x>)=0$ for $x,y,z \in M$

My question is how does this generalize the case of Lie Algebra over a field? And what is the motivation behind the second condition? Why can't we simply define a Lie structure over a ring to be a alternating bilinear form which satisfies Jacobi identity?

Here's a link to the paper https://www.sciencedirect.com/science/article/pii/0021869369900866