Example for a faithful and finite dimensional representation over a sovable and finite dimensional Lie-Algebra

I am new to StackExchange. I am learning about Lie-Algebras and I was wondering whether somebody can give me an example for a finite dimensional and faithful representation of a sovable and finite dimensional Lie-Algebra $$\mathcal{L}$$ over a Vektorspace V (finite dimensional): $$\rho: \mathcal{L} \longrightarrow gl(V)$$, such that if $$\mathcal{N}$$ is the Nil-Radical of $$\mathcal{L}$$, $$~ \rho(\mathcal{N})$$ is nilpotent.

Such a representation should exists (see: Theorem 9 of https://terrytao.wordpress.com/2011/05/10/ados-theorem/). I can think of one for a nilpotent Lie-Algebra, but I have trouble finding one for a sovable Lie-Algebra.

Thank you! :)

• You are right, thanks. I misused the term "Nil-representation". I changed the question. Feb 26, 2021 at 13:13

What about the natural action of the Lie algebra of upper triangular $$n\times n$$ matrices on $$k^n$$? The nilradical consists of strictly upper triangular matrices, which are nilpotent.
• I thought "example for a finite-dimensional faithful representation of a (given) solvable Lie algebra". Then a minimal example in dimension $2$ already works. Feb 27, 2021 at 9:19
Let $$\mathfrak{r}_2(K)$$ be the non-abelian Lie algebra of dimension $$2$$ over $$K$$ with basis $$(e_1,e_2)$$ and $$[e_1,e_2]=e_1$$. Then the adjoint representation is a faithful linear representation of $$\mathfrak{r}_2(K)$$ into $$\mathfrak{gl}_2(K)$$ and the nilradical $$\mathfrak{n}=\langle e_1\rangle$$ is mapped to the nilpotent matrix $${\rm ad}(e_1)=\begin{pmatrix} 0 & 1 \cr 0 & 0 \end{pmatrix}.$$ The adjoint representation is faithful in this case, because its kernel, the center of the Lie algebra, is trivial. This is never the case for non-abelian nilpotent Lie algebras.