When can you build up all representations from the fundamental and antifundamental ones?

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?

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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 $V^{\ast}$.
Nope. I've mostly been learning my Lie theory in the wilderness. One mild warning: there is an obvious representation of $\mathfrak{so}(n)$ which is not a faithful representation of the corresponding simply-connected compact Lie group when $n \ge 3$ (namely $\text{Spin}(n)$), so the hypotheses of this result don't apply. Indeed, you can't get spin representations from it. –  Qiaochu Yuan Feb 21 '13 at 21:01