Where I can find classifications of Lie algebra $A_{4.7}^{-1}$?

Question

My Lie algebra with commutation relation

$[e_2, e_3] = e_3,\;[e_2, e_4] = -e_4,\;[e_3, e_4] = -e_1$

is isomorphic to Lie algebra $A_{4.7}^{-1}$ through transformations

$e_1\mapsto e_1,\;e_2\mapsto - e_4,\;e_3\mapsto e_3,\;e_4\mapsto e_2$

I tried to find classification of algebra $A_{4.7}^{-1}$ in Patera and Winternitz but there I find only classification of $A_{4.7}$.

Can anybody please suggest me article where classification of $A_{4.7}^{-1}$ is given ?

Answer 1

Your Lie algebra is exactly $A_{4,8}^{-1}$ in Roman Popovych's classification, page $17$, Table $5$, after the suggested base change. The Lie brackets are given by $$ [x_2,x_3]=x_1,\;[x_2,x_4]=x_2,\;[x_3,x_4]=-x_3. $$ So Roman Popovych is right. There only was a typo in the index, it seems.