Exercise 15, § 5

Let $K$ be a field of characteristic 2. The identity representation of the nilpotent Lie algebra $\mathfrak{sl}(2, K)$ on $K^2$ defines a semi-direct product $h$ of $\mathfrak{sl}(2, K)$ by $K^2$. Show that $h$ is solvable but that $\mathfrak{D}h$ is not nilpotent. Deduce that $h$ admits no faithful linear representation by triangular matrices. Show also that the conclusions of Exercise 5 are false.