Prime-dimensional faithful irreducible module forces simplicity ($\dim V\text{ prime},V\text{ faithful irreducible}⟹L\text{ simple}$)

Let a semisimple complex Lie algebra $L={⨁}_i{L}_i$ act faithfully and irreducibly on a space $V$ of prime dimension. Complete reducibility for $L_1$ splits $V$ into isotypic components. The commuting other ideals preserve these components, so irreducibility leaves just one, of the form $W_1\otimes {\mathrm{Hom}}_{L_1}(W_1,V)$. Repeating with the other factors gives $V\cong {⨂}_i{W}_i$, with each $W_i$ an irreducible module for its simple ideal.

Faithfulness makes every factor nontrivial. A nonabelian simple Lie algebra has only the trivial one-dimensional module, so $\dim W_i\ge 2$. Their product is the prime $\dim V$, forcing just one simple ideal. Hence $L$ is simple. This is a representation-dimension obstruction, independent of the classification of root systems.