My query is related to infinite Lie algebra of KP equation having commutation relations are given by
$[X(f_1), X(f_2)] = X(f_1\dot{f_{2}}-f_2\dot{f_{1}})$
$[X(f), Y(g)] = Y(f\dot{g}-\frac{2}{3}\dot{f}g)$
$[X(f), Z(h)] = Z(f\dot{h}-\frac{1}{3}\dot{f}h)$
$[Y(g_1), Y(g_2)] = \frac{2}{3}\sigma Z(\dot{g_1}g_{2}-g_1\dot{g_{2}})$
$[Y(g), Z(h)] = 0$
$[Z(h_1), Z(h_2)] = 0$ here the dots are derivatives with respect to $t$.
In the article, it is proved that the infinite Lie algebra $\left\{{X(f), Y(g), Z(h)}\right\}$ can be written as Levi decomposition of $\left\{{X(f)}\right\}$ and $\left\{{ Y(g), Z(h)}\right\}$ here later is solvable ideal.
How I can prove this Levi decomposition ?
