Exercise 6, § 6

Let g be a Lie algebra and D∞g the intersection of the D^p g (p = 1, 2, ...). Show that g/D∞g is solvable and that D(D∞g) = D∞g. For g to be isomorphic to the product of a semi-simple algebra and a solvable algebra, it is necessary and sufficient that D∞g be semi-simple.