  In this work, we consider Lie algebras L containing a subalgebra isomorphic
to sl3 and such that L decomposes as a module for that sl3 subalgebra into
copies of the adjoint module, the natural 3-dimensional module and its dual,
and the trivial one-dimensional module. We determine the multiplication in L
and establish connections with structurable algebras by exploiting symmetry
relative to the symmetric group S4.
