Generalization of the almost-representation theorem to real compact Lie algebras

Let $\mathfrak{g}$ be a real compact Lie algebra, and replace the operators and relations in Theorem

by the corresponding almost-representation data for $\mathfrak{g}$. In particular, replace the relation on the \left-hand side of (R1) by the relation determined by the Casimir element of $\mathfrak{g}$. **Generalization conjecture.** The analogue of Theorem

holds for all real compact Lie algebras. The theorem for $\mathfrak{su}(2)$ establishes the model case: sufficiently accurate almost representations satisfying the corresponding dimension bound are forced to have the expected dimension and to lie within bounded operator-norm distance of an irreducible representation. Whether the same conclusion holds for every real compact Lie algebra is left as a generalization.