Suppose $X/k$ is a finite commutative group scheme over a perfect field. Then we know that the category $\mathcal{N}$ of finite commutative group schemes over $k$ is abelian and isomorphic to a direct product $$\mathcal{N} = \mathcal{N}_{loc} \times \mathcal{N}_{red} = \mathcal{N}_{ll} \times \mathcal{N}_{lr} \times \mathcal{N}_{rr} \times \mathcal{N}_{rl}$$ where $\mathcal{N}_{loc}$, $\mathcal{N}_{red}$ denote the full subcategories of $\mathcal{N}$ corresponding to objects whose corresponding rings are either reduced or local respectively. One can describe this at the level of bialgebras, apparently it corresponds to a "direct sum" of bialgebras.
$\textbf{Question:}$ Given a finite commutative group scheme $G$ one can of course calculate the reduced and local components, but how do you determine the local-local, local-reduced etc., components? It's not clear to me if this is easy from formal properties of duality. Is it true that $G \cong G_{ll} \times_k G_{rr} \times_k G_{lr} \times_k G_{rr}$.