Faithful representation of the Heisenberg group
Let $\mathfrak{h}_m$ be a $2m+1$-dimensional Heisenberg Lie algebra, with basis $e_1,\ldots e_m,f_1,\cdots ,f_m,z$ and Lie brackets $[e_i,f_i]=z$ for all $i$. One can show that the minimal dimension of a faithful representation of $\mathfrak{h}_m$ is $\mu(\mathfrak{h}_m)=m+2$ - see here, Lemma $1$. For the $3$-dimensional Heisenberg Lie algebra this says $\mu(\mathfrak{h}_1)=3$. Hence there is no $2$-dimensional faithful representation $\rho \colon\mathfrak{h}_1\rightarrow \mathfrak{gl}_2(\mathbb{R})$.