The problem I faced is how to organize a set of finite-dimensional irreducible representations $U_\alpha$ of some simple Lie algebra $g$ into an Lie algebra $A$ that contains $g$ as a Lie subalgebra and such that $A$ as a $g$-module decomposes into $\bigoplus_\alpha U_\alpha$. The set of irreps $U_\alpha$ of $g$ is the input data (the adjoint of $g$ can be thought of as one of $U_\alpha$).

A simple positive answer is given by the universal enveloping algebra evaluated at some other irreducible representation, say $V$, then $A=U(g)|_{V}\sim V\otimes V^* $ is an algebra and $A$ as a $g$-module is given simply by decomposing $V\otimes V^*=\bigoplus_\alpha U_\alpha$ into irreps $U_\alpha$. So given $U_\alpha$ that come out of $V\otimes V^*$ there exists an algebra $A$, whose 'generators' are given by $U_\alpha$. This gives a lot of algebras of this type. I wonder if this is the complete answer. This is reminiscent of the fusion problem in conformal field theory.

Fusion categories arising from semisimple Lie algebras, Comm. Math. Physics 169 (1995), 563-588 (see their section 1 and references). Maybe that's too far from your question; it's also technical. $\endgroup$