Let $g$ be a semi-simple Lie algebra, and $0 \to I \to h \to g \to 0$ an Abelian extension of $g$. Then $g$ acts on $I$. Considering $g$ under the adjoint action, when is there a $g$-module isomorphism between $g$ and the k-th exterior power $\Lambda^k(I)$ for some $k$? Only when $g = so(n)$, $k=2$, $k = n-2$?