Then the map \Phi _0 is well-defined for all k. Indeed, \Phi _0 is defined by \begin{aligned}&e_1 \mapsto -\gamma \textrm{e}^c,\quad e_2 \mapsto \beta ,\quad e_3 \mapsto \textrm{e}^c,\\&h_1 \mapsto \displaystyle -2J+:\!\gamma \beta \!:-\frac{2k+9}{6}c+...