For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.

More specifically, can somebody point me to a table detailing, for each series, those dominant weights for faithfulness fails.

**Edit**: To make my question clearer, I am asking in relation to the answer of this question, which asks when can you build up all representations from the fundamental and antifundamental ones? It is answered that, for an irreducible Lie algebra representation $V$, when the corresponding representation of $G$ is faithful, any other irreducible Lie algebra can be found in a tensor product $V^{\otimes k}$, for sufficiently high $k$. The answer is qualified with the following comment:

One mild warning: there is an obvious representation of $\frak{𝔰𝔬}(𝑛)$ which is not a faithful representation of the corresponding simply-connected compact Lie group when $n\geq 3$

Is the "obvious representation" of $\frak{so}(n)$ the only representation for which this happens, or are there others?