  We construct ternary self-distributive (TSD) objects from compositions of
binary Lie algebras, $3$-Lie algebras and, in particular, ternary Nambu-Lie
algebras. We show that the structures obtained satisfy an invertibility
property resembling that of racks. We prove that these structures give rise to
Yang-Baxter operators in the tensor product of the base vector space and, upon
defining suitable twisting isomorphisms, we obtain representations of the
infinite (framed) braid group. We consider examples for low-dimensional Lie
algebras, where the ternary bracket is defined by composition of the binary
ones, along with simple $3$-Lie algebras. We show that the Yang-Baxter
operators obtained are not gauge equivalent to the transposition operator, and
we consider the problem of deforming the operators to obtain new solutions to
the Yang-Baxter equation. We discuss the applications of this deformation
procedure to the construction of (framed) link invariants.
