Exercise 14, § 11

Assume that $\mathfrak{g}$ is semi-simple. Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{g}$, and let $f_1$ and $f_2$ be two homomorphisms from $\mathfrak{g}$ to a finite dimensional Lie algebra. Show that $f_1|\mathfrak{p}=f_2|\mathfrak{p}$ implies that $f_1=f_2$. (Reduce to the case in which $\mathfrak{g}$ is split, then to the case in which $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k)$, and use Lemma 1 of no. 1.)