consider the Heisenberg group $Heis(\mathbb{R})$ and $\mathfrak{h}$ it's Lie algebra. Consider $C\in \mathfrak{h}$ given by
$$C=\begin{bmatrix} 0 && 0 && 1\\ 0 && 0 && 0\\ 0 && 0 && 0 \end{bmatrix}$$
Now the goal is to show that for any representation of Lie algebras $\rho : \mathfrak{h} \to \mathfrak{gl}(n,\mathbb{R})$ $\rho(C)$ is a nilpotent matrix. The proof is a bit tricky and the notes I read show that $\rho(C)$ has only $0$ eigenvalues to show that it is nilpotent. But can't we just say $\rho(C)^n=\rho(C^n)=\rho(0)=0$ for some $n$, since $\rho$ is a representation?