Patera and Winternitz have carried out extensive classification of three and four dimensional Lie algebras. When I tried to look for classification for three dimensional Lie algebra with non-zero commutations given as:
$[e_{1}, e_{2}]=e_{2}, [e_{1}, e_{3}]=2\,e_{3} $
I could not find corresponding classification in table 1, similarly for four dimensional Lie algebra with non-zero commutation given as:
$[e_{1}, e_{3}]=e_{3}, [e_{1}, e_{4}]=e_{4}, [e_{2}, e_{3}]=e_{3}$
there is not any classification given there in table 2. It seems that they have missed these three and four dimensional algebras, I guess this might be due large number of possible Lie algebras of dimension three and four and it is natural for them to miss those algebras.
I wonder is there any limit to number of three and four dimensional Lie algebras with different non-zero commutations?
Your 3-dimensional Lie-algebra is isomorphic to $A_{3,5}^{1/2}$ in the table: Denoting by $(\mathfrak g,[.,.])$ your Lie-algebra (with non-zero relations $[e_1,e_2] = e_2$ and $[e_1,e_3] = 2e_3$) and by $(A_{3,5}^{1/2},(.,.))$ the Lie-algebra with non-zero relations $(e_1,e_3) =e_1$ and $(e_2,e_3) = \frac12 e_2$, then $$ \phi\colon \mathfrak g\longrightarrow A_{3,5}^{1/2},\quad e_1\mapsto -2e_3,\; e_2\mapsto e_2,\; e_3\mapsto e_1 $$ gives an isomorphism of Lie-algebras: Namely \begin{align*} \phi([e_1,e_2]) &= \phi(e_2) = e_2 = (e_2,2e_3) = (-2e_3,e_2) = (\phi(e_1), \phi(e_2))\\ \phi([e_1,e_3]) &= \phi(2e_3) = 2e_1 = (e_1,2e_3) = (-2e_3,e_1) = (\phi(e_1),\phi(e_3))\\ \phi([e_2,e_3]) &= 0 = (e_2,e_1) = (\phi(e_2),\phi(e_3)), \end{align*} i. e. $\phi$ preserves the Lie-brackets.
Your 4-dimensional Lie-algebra is isomorphic to $2A_2$, as can be seen by replacing your $e_1$ by $e_1-e_2$ (the non-zero relation then become $[e_1,e_4]=e_4$, $[e_2,e_3] = e_3$). I leave it to you, to write down the explicit isomorphism.