Here is a construction of a very broad class of "Lie-like" algebras, and I want to know more about them.

Here is the main definition: Suppose $\mathfrak{g}$ is a complex semsimple Lie algebra and $\Gamma$ is a finite abelian group. Define a "hybrid algebra" over $(\mathfrak{g},\Gamma)$ as a pair $(V,\Phi)$ where $V$ is a collection of $\mathfrak{g}$-modules $V_a$ indexed by $\Gamma$ and $\Phi$ is a collection of intertwining maps $\phi_{a,b}:V_a\otimes V_b\rightarrow V_{a+b}$ indexed by $\Gamma\times\Gamma$.

What is known about these things? Here are a few things that I know. This is a generalization which encompasses semisimple Lie algebras and classical Lie superalgebras. Thus, for example, a Lie superalgebra $\mathfrak{g}$ is a particular type of hybrid algebra over the pair $(\mathfrak{g}_0,\mathbb{Z}/2\mathbb{Z})$, where $\mathfrak{g}_0$ is the even component of $\mathfrak{g}$. Similarly, one may realize the exceptional $52$-dimensional Lie algebra $\mathfrak{f}_4$ as a hybrid algebra over the pair $(\mathfrak{d}_4,V)$, where $V$ is the Klein four group and using the triality representations of $\mathfrak{d}_4\cong\mathfrak{so(8)}$. There are similar (and similarly elegant) constructions for other semisimple Lie algebras.

Beyond Lie algebras and Lie superalgebras, what are the interesting classes of hybrid algebras? What else can we say?