I have the following definition of an L-module
We say that V is an L-module if there is a k-bilinear mapping L × V → V sending a pair (x, v) ∈ L × V to x.v ∈ V such that [x, y].v = x.(y.v) − y.(x.v) for all v ∈ V and x, y ∈ L.
My confusion arises with the square bracket. I was taught the square bracket is used to denote the product in L. However, I was also taught about the commutator product [a,b] = a.b - b.a which is used to turn associative algebras into L algebras. Does the square bracket take this definition in the definition of an L-module? I get confused when the square brackets are used in my notes because I'm never sure if they are referring to a general product in a Lie Algebra or the commutator defined above. I just started studying Lie Algebras and this has been confusing me for a while. Thanks
In general, the bracket $[x,y]$ for an abstract Lie algebra $L$ denotes a Lie bracket, i.e., a skew-symmetric, bilinear product satisfying the Jacobi identity. For the general linear Lie algebra $L=\mathfrak{gl}_n(K)$ we have a special Lie bracket $[A,B]=AB-BA$, given by the commutator of matrices. For general definitions, like an $L$-module above, we always assume that $[x,y]$ is an abstract Lie bracket. However, there is a famous theorem of Ado and Iwasawa, that we may represent any finite-dimensional Lie algebra by matrices, together with the special Lie bracket given by the commutator.