Is there a name for an associative algebra which is further endowed with a Lie algebra structure such that the Leibniz rule holds, i.e., $$[a, b \circ c]=[a,b]\circ c+b\circ [a,c], \hspace{15mm}(*)$$ 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, although the notion itself seems fairly natural.

Indeed, any associative algebra is readily endowed with a Lie algebra structure with the commutator $[a,b]=a\circ b-b\circ a$ which obeys the Leibniz rule $(*)$ but there could exist other Lie algebra structures which still obey $(*)$.

Any relevant references are greatly appreciated. Thanks in advance.