Exercise 2, § 2

Suppose that $\mathbf{K}$ is a field. Let $g$ be a Lie $\mathbf{K}$-algebra and $U$ its enveloping algebra. (a) Let $u, v$ be in $U$. If $u$ is of filtration $> n$, and $v$ of filtration $> p$, then $uv$ is of filtration $> n + p$ (use Theorem 1). (b) Deduce from (a) that the only invertible elements of $U$ are the scalars. (c) Deduce from (b) that the Jacobson radical of $U$ is zero.