Schur multiplier of a Lie ring

1 second homology group It is the second homology group for trivial Lie ring action $\! H_2(L;\mathbb{Z})$.
2 second cohomology group (?) It is the second cohomology group for trivial Lie ring action of $L$ on $\mathbb{C}^*$, i.e., the group $\! H^2(L,\mathbb{C}^*)$. Note that we could replace $\mathbb{C}^*$ by any divisible abelian group viewed as an abelian Lie ring.
3 Baer invariant It is the Baer invariant of $G$ corresponding to the subvariety of abelian Lie rings in the variety of Lie rings.
4 Kernel of Lie bracket It is the kernel of the Lie bracket homomorphism from exterior square to derived subgroup, i.e., it is the kernel of the map $L \wedge L \to [L,L]$ given by $x \wedge y \mapsto [x,y]$. Here $L \wedge L$ is the exterior square of $L$ and $[L,L]$ is the derived subring of $L$.
5 Hopf's formula It is given by Hopf's formula for Schur multiplier: If $L$ is isomorphic to the quotient of a free Lie ring $F$ by an ideal $R$, then $M(G) = (R \cap [F,F])/[R,F]$.