How is the representations of this type algebras

Let V be a vector space, X=(X1,...,Xn) sets that determine inner derivations

adX1...Xn. A representation ρ of a FA on V is a multilinear map ρ: X a ρ(X1,....,Xn) such that:

[ρ(X),ρ(Y)]=ρ(X.Y)

ρ(X1,.....,Xn-2,[Y1,....,Yn])=\sum((-1)^n-i)ρ(Y1,...,Y'i,...,Yn)ρ(X1,...,Xn-2,Yi)

this is the representation in the sense of Kasymov

We need an integral bilinear form on integral weight systrm for any representation.

https://www.researchgate.net/publication/234849476_Realizations_of_3-Lie_algebras

Article Realizations of 3-Lie algebras