Semisimplifying Lie algebras of $J$-ternary algebras in characteristic $3$

We describe a class of Lie superalgebras in characteristic $3$, containing the Elduque-Cunha superalgebras $\mathfrak{g}(3,3), \mathfrak{g}(6,6)$ and the Elduque superalgebra $\mathfrak{el}(5,3)$, using the tensor product of composition algebras. For the Lie superalgebra $\mathfrak{el}(5,3)$, this allows us to move beyond the contragredient construction and it also allows us to construct more general forms. We also describe how one obtains these Lie superalgebras using the semisimplification functor on the representation category $\mathbf{Rep}(\alpha_3)$ to Lie algebras of type $E_6, E_7$ and $E_8$, in line with how Arun Kannan applied this functor to the split algebras. We further apply this functor more broadly to the class of Lie algebras coming from $J$-ternary algebras over fields of characteristic $3$.