Lie algebras with prescribed $\mathfrak{sl}_3$ decomposition

In this work, we consider Lie algebras $\mathcal {L}$ containing a subalgebra isomorphic to $\mathfrak{sl}_3$ and such that $\mathcal {L}$ decomposes as a module for that $\mathfrak{sl}_3$ subalgebra into copies of the adjoint module, the natural three-dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in $\mathcal {L}$ and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group $\mathsf {S}_4$.