Return to Revisions

I did not know these identities but after a small search, i think that from a formal point of view, some relations are missing from your post:
In: http://mathworld.wolfram.com/KaehlerIdentities.html
some additional relations (eqs (10)-(13)) are included; if you take into account these additional relations, then your algebra reasonably seems to be a Lie superalgebra with a basis for the even subspace consisting of $L$, $\Lambda$ and a basis for the odd subspace consisting of $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$,