Corollary

Let $\mathfrak{h}$ be a nilpotent Lie algebra, and M a finite dimensional $\mathfrak{h}$-module such that $M^0(\mathfrak{h}) = 0$. Let $f:\mathfrak{h}\rightarrow M$ be a linear map such that

$f([x, y]) =x.f(y)-y.f(x)$ for $x, y\in \mathfrak{h}$.

There exists $a\in M$ such that $f(x) =x.a$ for all $x\in \mathfrak{h}$.

Let $N = M\times k$. Make $\mathfrak{h}$ operate on N by the formula

The identity satisfied by $f$ implies that N is an $\mathfrak{h}$-module (Chap. I, §1, no. 8, Example 2). The map $(m, \lambda ) \rightarrow \lambda$ from N to $k$ is a homomorphism from N to the trivial $\mathfrak{h}$-module $k$. By Prop. 9 (iv), it follows that $N^0(\mathfrak{h})$ contains an element of the form $(a,1)$ with $a\in M$. In view of the hypothesis on M,

so $N^0(\mathfrak{h})$ is of dimension 1 and hence is annihilated by $\mathfrak{h}$. Thus, $x.a-f(x) = 0$ for all $x\in \mathfrak{h}$, which proves the corollary.