The realizations (of an algebra through another algebra) you are speaking about are actually homomorphisms. And as such they should map between algebraic structures of the same kind: that is from algebras to algebras, from Lie alg to Lie alg, from graded algebras to graded algebras (graded by the same group), etc
Since you are considering Lie superalgebras and realize them through boson-fermion operators, what you are actually doing is to consider the algebra mixing the bosonic/fermionic degrees of freedom as a superlagebra: this means that you consider it eqquiped with a $\mathbb{Z}_2$-grading and you pick the unique color function available (unique because there is ony a single bicharacter of the $\mathbb{Z}_2$ group). This is the function: $\theta:\mathbb{Z}_2\times\mathbb{Z}_2\rightarrow\mathbb{C}^*$ explicitly given by $\theta(a,b)=(-1)^{\deg a\deg b}$, where $a,b\in\mathbb{Z}_2$ and $\deg=0,1$ depending on whether the corresponding element is even or odd. This bicharacter actually determines the exact form of the brackets in the LS; that is whether they are commutators ($\theta(a,b)=1$) or anticommutators ($\theta(a,b)=-1$).
If you go to more general gradings (that is $\mathbb{Z}_3$ which interests you -but it could even be an arbitrary finite, abelian group $\mathbb{G}$) then the "bracket" will generally be defined by $[a,b]=ab-\theta(a,b)ba$, where $\theta:\mathbb{G}\times\mathbb{G}\rightarrow\mathbb{C}^*$ so essentially the color function (take it as a synonym for a bicharacter in this setting) is what determines the form of the bracket (in higher grading groups with more bicharacters available, the bracket need not be an (anti)commutator, it can have a more general form). $\theta$ also determines the behaviour of the multiplication in the tensor product algebras, so we speak about braided, graded tensor products but this is possibly another story.
If you are interested in this point of view, you coud take a look (excuse me in advance for the self-citation but i think it is relevant here) at:
- Super-Hopf realizations of Lie Superalgebras: Braided Paraparticle extensions of the Jordan-Schwinger map
- Gradings, Braidings, Representations, Paraparticles: some open problems (see expecially the discussion of p. 80-81 and section 5.1)
The first paper mostly reviews the relevant notions while the second one utilizes realizations in a way which -i think- is close to what you are doing. The difference is that i am using (in this references) "bigger" algebras than the supersymmetric mixture of bosons and fermions you are considering. I am utilizing algebras which mix parabosonic and/or parafermionic generators (your algebras can be recovered as quotients of the later). This gives the opportunity to consider other grading groups (i am mainly focusing to the case of a $\mathbb{Z}_2\times\mathbb{Z}_2$ grading group and its colors or bicharacters if you prefer). It seems that such a line of investigation presents interest from both the mathematics and the physics point of view. In:
the author seems to pursue further similar ideas and methods (focusing more on the physics point of view) while i (together with a couple of colleagues) have some older work on further applications of the $\mathbb{Z}_2\times\mathbb{Z}_2$ (focusing more on th representation-theoretic point of view: that is on the construction of representations of Lie superalgebras through paraparticle realisations). I can send you the relevant references if you are interested in these.
I am not sure if the above shed some light in your questions and your research but i hope you'll find something of interest in these.