Which is the name for an associative algebra, which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c],$$ where $[,]$ is the Lie algebra commutator (Lie bracket) and $\circ$ is the multiplication in our algebra?

If the multiplication $\circ$ is commutative, then such an algebra is called a Poisson algebra but I was not able to find in the literature the name for the case when the commutativity assumption is dropped. Any relevant references are greatly appreciated.

Thanks in advance.