Anticommutative Algebras Satisfying Standard Identities of Degree Four

We define an anticommutative $\Phi$-algebra $A(D,a)$ whose multiplication generalizes the concept of a Jacobi bracket in the form (4). It is proved that $A(D,a)$ is a $J$-algebra and that it satisfies a standard identity of degree four. A subclass $\mathfrak M$ of algebras $A(D,a)$ over $\Phi$ which is connected with some class of 3-Lie algebras is distinguished. We establish a criterion of being simple for factor algebras of non-Lie algebras in $\mathfrak M$, given a 1-dimensional annihilator, and then use it to construct examples of simple infinite-dimensional (of dimension $p^3-1$) non-Lie $J$-algebras over a field $\Phi$ satisfying standard identities of degree 4, if the characteristic $p$ of $\Phi$ is zero (for $p>2$). Also, the criterion of algebras belonging to $\mathfrak M$ is given.