Homework Set #8

1. Prove that every maximal toral subalgebra of sl(2) is 1-dimensional.
2. Show that every semisimple Lie algebra of dimension less than 6 is isomorphic to sl(2).
3. Show that if L is a Lie algebra and I an abelian ideal, then (ad(x)ad(y))²=0 for all x∈I, y∈L. Deduce that if the Killing form of L is non-degenerate, then L is semisimple.
4. Prove that every root system of type A_n is the root system of a semisimple Lie algebra.