In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) 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 $x \otimes y \mapsto \langle x,y\rangle$, satisfying

• $\langle x,x\rangle = 0$;

• $\langle x,y\rangle t(uv-vu)=(xy-yx)t\langle u,v\rangle$, for all $x,y,u,v \in M$ and $t \in T(M)$; and

• $(\langle x,y\rangle z-z\langle x,y\rangle)+(\langle y,z\rangle x-x\langle y,z\rangle)+(\langle z,x\rangle y-y\langle z,x\rangle) = 0$, for $x,y,z \in M$.

My question is, how does this generalize the case of a 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 an alternating bilinear form which satisfies the Jacobi identity?

In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) 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 $x \otimes y \mapsto \langle x,y\rangle$, satisfying

• $\langle x,x\rangle = 0$;

• $\langle x,y\rangle t(uv-vu)=(xy-yx)t\langle u,v\rangle$, for all $x,y,u,v \in M$ and $t \in T(M)$; and

• $(\langle x,y\rangle z-z\langle x,y\rangle)+(\langle y,z\rangle x-x\langle y,z\rangle)+(\langle z,x\rangle y-y\langle z,x\rangle) = 0$, for $x,y,z \in M$.

My question is, how does this generalize the case of a 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 an alternating bilinear law which satisfies the Jacobi identity?

In Higgins's paper Baer invariants and the Birkhoff–Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) 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 $x \otimes y \mapsto \langle x,y\rangle$, satisfying

• $\langle x,x\rangle = 0$;

• $\langle x,y\rangle t(uv-vu)=(xy-yx)t\langle u,v\rangle$, for all $x,y,u,v \in M$ and $t \in T(M)$; and

• $(\langle x,y\rangle z-z\langle x,y\rangle)+(\langle y,z\rangle x-x\langle y,z\rangle)+(\langle z,x\rangle y-y\langle z,x\rangle) = 0$, for $x,y,z \in M$.

My question is, how does this generalize the case of a 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 the Jacobi identity?

In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) 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 $x \otimes y \mapsto \langle x,y\rangle$, satisfying

• $\langle x,x\rangle = 0$;

• $\langle x,y\rangle t(uv-vu)=(xy-yx)t\langle u,v\rangle$, for all $x,y,u,v \in M$ and $t \in T(M)$; and

• $(\langle x,y\rangle z-z\langle x,y\rangle)+(\langle y,z\rangle x-x\langle y,z\rangle)+(\langle z,x\rangle y-y\langle z,x\rangle) = 0$, for $x,y,z \in M$.

My question is, how does this generalize the case of a 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 an alternating bilinear form which satisfies the Jacobi identity?

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

In Higgins's paper Baer invariants and the Birkhoff–Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) 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 $x \otimes y \mapsto \langle x,y\rangle$, satisfying

• $\langle x,x\rangle = 0$;

• $\langle x,y\rangle t(uv-vu)=(xy-yx)t\langle u,v\rangle$, for all $x,y,u,v \in M$ and $t \in T(M)$; and

• $(\langle x,y\rangle z-z\langle x,y\rangle)+(\langle y,z\rangle x-x\langle y,z\rangle)+(\langle z,x\rangle y-y\langle z,x\rangle) = 0$, for $x,y,z \in M$.

My question is, how does this generalize the case of a 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 the Jacobi identity?