rt.representation theory - Isomorphism classes of nilpotent Lie algebras

I have found a counterexample while studying Lie algebra degenerations. Consider the following filiform nilpotent Lie algebra $mathfrak{g}$ of dimension $13$, given by the brackets with respect to a basis
$(e_1,ldots ,e_{13})$:
begin{align*}
[e_1,e_i] & = e_{i+1},quad 2le ile 12 \[0.2cm]
[e_2,e_3] & = e_5 \
[e_2,e_4] & = e_6 \
[e_2,e_5] & = frac{9}{10}e_7-e_9 \
[e_2,e_6] & = frac{4}{5}e_8-2e_{10} \
[e_2,e_7] & = frac{5}{7}e_9-frac{335}{126}e_{11}+ frac{22105}{15246}e_{13}\
[e_2,e_8] & = frac{9}{14}e_{10}-frac{125}{42}e_{12}\
[e_2,e_9] & = frac{9}{12}e_{11}-frac{4421}{1452}e_{13}\
[e_2,e_{10}] & = frac{8}{15}e_{12}\
[e_2,e_{11}] & = frac{27}{55}e_{13}\[0.5cm]
[e_3,e_4] & = frac{1}{10}e_{7}+e_{9}\
[e_3,e_5] & = frac{1}{10}e_{8}+e_{10}\
[e_3,e_6] & = frac{3}{35}e_9+frac{83}{126}e_{11}-frac{22105}{15246}e_{13}\
[e_3,e_7] & = frac{1}{14}e_{10}+frac{20}{63}e_{12}\
[e_3,e_8] & = frac{5}{84}e_{11}+frac{697}{10164}e_{13}\
[e_3,e_{9}] & = frac{1}{20}e_{12}\
[e_3,e_{10}] & = frac{7}{165}e_{13}\
[e_4,e_5] & = frac{1}{70}e_9+frac{43}{126}e_{11}+frac{22105}{15246}e_{13}\
[e_4,e_6] & = frac{1}{70}e_{10}+frac{43}{126}e_{12}\
[e_4,e_7] & = frac{1}{84}e_{11}+frac{7589}{30492}e_{13}\
[e_4,e_8] & = frac{1}{105}e_{12}\
[e_4,e_9] & = frac{1}{132}e_{13}\[0.5cm]
[e_5,e_6] & = frac{1}{420}e_{11}+frac{313}{3388}e_{13}\
[e_5,e_7] & = frac{1}{420}e_{12}\
[e_5,e_8] & = frac{3}{1540}e_{13}\[0.5cm]
[e_6,e_7] & = frac{1}{2310}e_{13}
end{align*}
Let $mathfrak{h}$ be the ideal generated by $e_3,ldots , e_{13}$ of codimension $2$. Then $mathfrak{g}/mathfrak{h}={ [e_1],[e_2]}$, and we
may write $L={ alpha[e_1]+beta[e_2]}$, where $alpha,beta$ vary over all complex numbers. Hence the preimage is $L(alpha,beta)={alpha e_1+beta e_2,e_3,ldots ,e_{13} }$, which is a $1$-codimensional ideal in $mathfrak{g}$. Now it is easy to see that
$$
L(1,alpha)simeq L(1,alpha') text{ if and only if } alpha=pm alpha'.
$$
So we obtain infinitely many non-isomorphic Lie algebras.