Exercise 8, § 2

Suppose that K is a field. Let l be a free Lie algebra with basic family $(\bar{x}_i)_{i \in I}$, let r be an ideal of l contained in $[l, l]$ and let g = $l/r$; let $x_i$ denote the image of $\bar{x}_i$ in g. Show the equivalence of the following properties: (i) g is a free Lie algebra. (ii) g admits $(x_i)$ as basic family (i.e. $r = \{0\}$). (iii) For every g-module M, $H^2(g, M) = \{0\}$. (iv) If g operates trivially on K, then $H^2(g, K) = \{0\}$. (The implications (ii) \Rightarrow (i) and (iii) \Rightarrow (iv) are obvious and (i) \Rightarrow (iii) follows from Exercise 7. To prove that (iv) \Rightarrow (ii) it suffices to show that $r = [l, r]$, cf. Exercise 5; otherwise take a hyperplane h of r containing $[l, r]$ and note that the extension $r/h \to l/h \to l/r = g$ is essential.)