Exercise 9, § 2

Let $g = \bigoplus_{n \geq 1} g_n$ be a graded Lie algebra. If h is a graded subalgebra of g such that $g = h + [g, g]$, show that $h = g$. If $(x_i)$ is a family of homogeneous elements of g, deduce that $(x_i)$ is a generating family if and only if the images of the $x_i$ in $g/[g, g]$ generate $g/[g, g]$. If K is a field on which g operates trivially and $H^2(g, K) = \{0\}$, show that the family $(x_i)$ is basic if and only if the images of the $x_i$ in $g/[g, g]$ form a basis of this vector space. (Use Exercise 8.)†