Exercise 9, § 1

Let $\mathfrak{g}$ be a Lie algebra in which $[[x, y], y] = 0$ for all $x$ and $y$. (a) Show that $3[[x, y], z] = 0$ for all $x, y, z$ in $\mathfrak{g}$. (Observe that $[[x, y], z]$ is an alternating trilinear function of $(x, y, z)$ and apply the Jacobi identity.) (b) Show similarly, using (a) and Exercise 8, that $[[[x, y], z], t] = 0$ for all $x, y, z, t$ in $\mathfrak{g}$.