What if replacing the anticommutativity by commutativity in the definition of Lie algebra?

Question

As the definition of Lie algebra in Wikipedia. A Lie algebra is a vector space together with a bilinear map, called Lie bracket, satisfying the alternativity and the Jacobi identity. The alternativity can be replaced by anticommutativity since they are equivalent under the Jacobi identity bilinearity.

I know the motivation of this definition. But what's going to happen if we replace the alternativity or anticommutativity by commutativity in the definition? Are there some mathematical notions related to this? I just wonder. Maybe it makes no sense at all.

Does anyone know about this? TIA...

Answer 1

We have an article about such algebras, which we called Jacobi-Jordan algebras, because they turn out to be Jordan algebras then in addition. The paper is

Jacobi-Jordan Algebras.

An algebra $A$ over a field $K$ is called a Jacobi-Jordan algebra if it satisfies the following two identities, \begin{align*} x\cdot y -y\cdot x & = 0 \label{1}\\ x\cdot (y\cdot z)+y\cdot (z\cdot x)+z\cdot (x\cdot y) & = 0 \label{2} \end{align*} for all $x,y,z\in A$.

Answer 2

There is a notion of super Lie algebra which is defined on graded vector space, the commutator of degree one is anticommutative

https://en.wikipedia.org/wiki/Lie_superalgebra