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 associative algebra $A$ that contains $g$ as a Lie subalgebra under commutators 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.
