Exercise 5, § 3

a) Show that there exists a Lie algebra $\mathfrak{g}$ over $k$ with basis $(x, y, z, t)$ such that

Show that $\mathfrak{g}$ is solvable and that $\mathfrak{k}=kx+ky+kz$ is a subalgebra of $\mathfrak{g}$.

b) Show that the elementary automorphisms of $\mathfrak{g}$ are the maps of the form $1 +\lambda$ ad$_{\mathfrak{g}}y+\mu$ ad$_{\mathfrak{g}}t$ where $\lambda , \mu\in k$.

c) Show that 1 + ad$_{\mathfrak{k}}x$ is an elementary automorphism of $\mathfrak{k}$ which does not extend to an elementary automorphism of $\mathfrak{g}$.

d) Let $\mathfrak{s}$ be a semi-simple subalgebra of a Lie algebra $\mathfrak{a}$. Show that every elementary automorphism of $\mathfrak{s}$ extends to an elementary automorphism of $\mathfrak{a}$.