If $\mathfrak{g}$ is a semisimple Lie algebra, is it true that $$ [\mathfrak{g}, \mathfrak{g}] = \mathfrak{g}.$$
Obviously the derived ideals must stabilize somewhere, but I don't see why $D^{1}(\mathfrak{g}) = \mathfrak{g}$. It seemed my professor was stating this result as a consequence of Weyl's Theorem, but I am not sure why this should follow (or even if this is true).
Yes, it is true. A semisimple Lie algebre is a direct sum $\bigoplus_{j=1}^n\mathfrak g_j$ of simple Lie algebras. And $[\mathfrak g_j,\mathfrak g_j]$ is an ideal of $\mathfrak g_j$. Since $\mathfrak g_j$ is simple, this ideal can only be $\mathfrak g_j$ itself. So$$[\mathfrak g,\mathfrak g]=\bigoplus_{j=1}^n[\mathfrak g_j,\mathfrak g_j]=\bigoplus_{j=1}^n\mathfrak g_j=\mathfrak g.$$