On the algebra generated by $\bar{\mu },\bar{\mathrm{\partial }},\mathrm{\partial },\mu$

In this note, we determine the structure of the associative algebra generated by the differential operators $\bar{\mu },\bar{\mathrm{\partial }},\mathrm{\partial },\mu$ that act on complex-valued differential forms of almost complex manifolds. This is done by showing it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential $[d,-]$, as well as its cohomology with respect to all its inner differentials.