Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I think this is true for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{C})$, or at least that's what physics books lead me to believe!
Is it true in general for semisimple Lie algebras?
Theorem: Let $G$ be a compact Lie group and let $V$ be a faithful (finite-dimensional, continuous, complex) representation of $G$. Then every (finite-dimensional, continuous, complex) irreducible representation of $G$ is a subrepresentation of a tensor product of copies of $V$ and $V^{\ast}$.
Proof. This follows from Stone-Weierstrass and the orthogonality relations for matrix coefficients. See Proof 3 in this blog post.
This gives the desired result for Lie algebras of simply-connected compact Lie groups (provided that "fundamental representation" implies "faithful representation of the corresponding simply-connected compact Lie group"; I am not actually sure what this term means).