Exercise 7, § 1

Let $\mathfrak{g}$ be a nilpotent Lie algebra and V a $\mathfrak{g}$-module such that $V^0(\mathfrak{g}) = 0$. Show that $H^p(\mathfrak{g},V) = 0$ for all $p\geq 0$. (Reduce to the case where V = $V^{\lambda}(\mathfrak{g})$, with $\lambda \not= 0$ and choose an element $x\in \mathfrak{g}$ such that $\lambda (x)\not= 0$. Apply Exerc. $6b$), remarking that the $\lambda_i$ are all zero and the $\mu_j$ are all equal to $\lambda (x).)$

Recover the Cor. of Prop. 9 (take $p= 1$)$.^4$