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

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.

• To make sure I understand you correctly: You wonder about non-associative commutative algebras which fulfill the Jacobi identity? – Peter Wildemann Feb 27 '18 at 19:13
• @PeterWildemann Absolutely! The non-associative commutative algebras which fulfill the Jacobi identity... – Q. Huang Feb 27 '18 at 19:32

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 $x,y,z\in A$.