Like with groups, the Schur multiplier of a Lie algebra can be defined in several ways, depending on the context and the specific definitions used. Here are three commonly used definitions:
$(1)$ Universal Central Extension: The Schur multiplier can be defined as the second cohomology group $H^2(\mathfrak{g}, \mathbb{C}^*)$ of the Lie algebra $\mathfrak{g}$ with coefficients in the multiplicative group $\mathbb{C}^*$ of complex numbers excluding zero. This definition relates to the universal central extension of $\mathfrak{g}$.
$(2)$ Commutator Kernel: The Schur multiplier can also be defined as the kernel of the natural map from the second exterior power of the Lie algebra $\mathfrak{g}$ to its associated graded Lie algebra. In this context, the Schur multiplier captures the commutativity obstruction in the Lie algebra.
$(3)$ Representations: Another way to define the Schur multiplier is in terms of representations. It can be characterized as the kernel of the natural map from the second exterior power of the adjoint representation of $\mathfrak{g}$ to the derived subalgebra.
These are the most commonly used definitions, but I am not getting the construction of the second and third definitions. If possible, please provide the reference.