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I did not know these identities but after a small search, i think that some relations are missing from your post:
In: http://mathworld.wolfram.com/KaehlerIdentities.html
some additional relations (see eqs (17)-(19) and 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 (or: $\mathbb{Z}_2$-graded lie algebra), with a basis of the even subspace consisting of $L$, $\Lambda$, $H$ and a basis of the odd subspace consisting of $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$.

Related:
Is the SUSY Algebra isomorphic for all Kähler Manifolds?

I did not know these identities but after a small search, i think that some relations are missing from your post:
In: http://mathworld.wolfram.com/KaehlerIdentities.html
some additional relations (see eqs (17)-(19) and 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 is a Lie superalgebra (or: $\mathbb{Z}_2$-graded lie algebra), with a basis of the even subspace consisting of $L$, $\Lambda$, $H$ and a basis of the odd subspace consisting of $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$.

Related:
Is the SUSY Algebra isomorphic for all Kähler Manifolds?

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 eqs (17)-(19) and 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 (or: $\mathbb{Z}_2$-graded lie algebra), with a basis of the even subspace consisting of $L$, $\Lambda$, $H$ and a basis of the odd subspace consisting of $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$.

Related:
Is the SUSY Algebra isomorphic for all Kähler Manifolds?

I did not know these identities but after a small search, i think that some relations are missing from your post:
In: http://mathworld.wolfram.com/KaehlerIdentities.html
some additional relations (see eqs (17)-(19) and 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 (or: $\mathbb{Z}_2$-graded lie algebra), with a basis of the even subspace consisting of $L$, $\Lambda$, $H$ and a basis of the odd subspace consisting of $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$.

Related:
Is the SUSY Algebra isomorphic for all Kähler Manifolds?

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 eqs (17)-(19) and 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 of the even subspace consisting of $L$, $\Lambda$, $H$ and a basis of the odd subspace consisting of $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$.

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 eqs (17)-(19) and 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 (or: $\mathbb{Z}_2$-graded lie algebra), with a basis of the even subspace consisting of $L$, $\Lambda$, $H$ and a basis of the odd subspace consisting of $\partial,\overline{\partial}, \partial^*,\overline{\partial}^*$.

Related:
Is the SUSY Algebra isomorphic for all Kähler Manifolds?