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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 (see the LHS of eqs (10)-(13); the RHS do not seem to be independent relations, they can probably be extracted from the rest) 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 $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$,