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The Lie algebra is filiform nilpotent and is generated by $e_1$ and $e_2$. It is known that any faithful Lie algebra representation $\rho:\mathfrak{f}_n\rightarrow \mathbb{gl}(V)$ of a $n$-dimensional filiform Lie algebra $\mathfrak{f}_n$ is of degree at least $n$. In the above example, we do not need to invoke Ado's theorem. It is enough to construct a faithful $5$-dimensional Lie algebra representation. By Lie's theorem, we assume that the operators $\rho(e_i)$ are simultaneously upper-triangular. We may even assume that $\rho(e_1)$ has canonical Jordan form. Then it is easy to compute explicit matrices $\rho(e_1)$ and $\rho(e_2)$, which determine the faithful representation. For details see, e.g., here.