Exercise 18, § 4

Let g be a 7-dimensional vector space over K with a basis $(e_i)_{1 \leq i \leq 7}$. An alternating bracket is defined on g by the formulae:

(1)

and all the other brackets $[e_i, e_j]$ are zero for $i < j$.

(a) For the Jacobi identity to hold, it is necessary and sufficient that:

(2)

(3)

(b) Suppose henceforth that all the $\alpha_{ij}$ are $\neq 0$. Show that the ideals $\mathscr{C}^2g, \mathscr{C}^3g, \mathscr{C}^4g, \mathscr{C}^5g, \mathscr{C}^6g$ admit the following bases: $(e_3, e_4, e_5, e_6, e_7)$, $(e_4, e_5, e_6, e_7)$, $(e_5, e_6, e_7)$, $(e_6, e_7)$, $(e_7)$. Show that the centralizer $h$ of $\mathscr{C}^6g$ admits the basis $(e_2, e_3, e_4, e_5, e_6, e_7)$.

(c) Let $(e'_i)_{1 \leq i \leq 7}$ be another basis of g such that $\mathscr{C}^6g$ admits the basis $(e'_7)$, $\mathscr{C}^5g$ admits the basis $(e'_6, e'_7)$, $\ldots$, $h$ admits the basis $(e'_2, e'_3, e'_4, e'_5, e'_6, e'_7)$. Show that

where

(d) Deduce that there exists, if K is infinite, an infinity of non-isomorphic nilpotent Lie algebras of dimension 7 over K and that there exist complex nilpotent Lie algebras which cannot be derived from real nilpotent Lie algebras by extending the base field from $\mathbf{R}$ to $\mathbf{C}$.