Theorem 3 (Ado)

Let $g$ be a Lie algebra and $n$ its largest nilpotent ideal. There exists a finite-dimensional faithful representation $\varphi$ of $g$ such that every element of $\varphi(n)$ is nilpotent.

The 1-dimensional Lie algebra $K$ admits finite-dimensional faithful representations $\tau$ such that every element of $\tau(K)$ is nilpotent, for example the representation

It is easily deduced that the centre $c$ of $g$, which is a product of 1-dimensional algebras, admits a finite-dimensional faithful representation $\sigma$ such that every element of $\sigma(c)$ is nilpotent. Let $\sigma_1$ be a finite-dimensional extension of $\sigma$ to $g$ such that every element of $\sigma_1(n)$ is nilpotent (Proposition 1); if $k$ denotes the kernel of $\sigma_1$, then $k \cap c = \{0\}$. On the other hand, let $\sigma_2$ be the adjoint representation of $g$, whose kernel is $c$; every element of $\sigma_2(n)$ is nilpotent. The direct sum $\rho$ of $\sigma_1$ and $\sigma_2$ is finite-dimensional; every element of $\rho(n)$ is nilpotent; and the kernel of $\rho$, contained in $k$ and in $c$, is zero, so that $\rho$ is faithful.