Exercise 3, § 2

Suppose that $\mathbf{K}$ is a field. Let $g$ be a Lie $\mathbf{K}$-algebra and $U$ its enveloping algebra. The left regular representation of $U$ corresponds to a representation $\rho$ of $g$ on the space $U$. Show that the set $U_+$ of elements of $U$ with no constant term is stable under $\rho$ and that $U_+$ has no supplement in $U$ stable under $\rho$. In particular, $\rho$ is not semi-simple. (This will be considered again in Theorem 2 of § 6.)