Let $g$ be a lie algebra, and $V$ be a vector space over $F.$ Assume that $\operatorname{char} (F) =0,$ and $\dim(V)=n>1.$ I want to show that $(\rho, V)$ be in irreducible representation of $g$ iff $(\rho^*, V^*) $ is an irreducible representation of $g$. Note that here $\rho^*$ is the contagradient representation of $\rho,$ and its defined as $\rho^*(X)(f) = -f(\rho(X)$ for any functional $f \in V^*.$
I was able to prove that if $(\rho^*, V^*)$ is irreducible, then so is $(\rho, V).$ I proved it using contradiction. I spent a fair amount of time to do the other direction, but I am completely stuck. Could you help me with the other direction? Thanks so much.