Let $${𝒪}_1$$ and $${𝒪}_2$$ be adjoint nilpotent orbits in a real semisimple Lie algebra. Write $${𝒪}_1$$ ≥ $${𝒪}_2$$ if $${𝒪}_2$$ is contained in the closure of $${𝒪}_1$$ . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E 8. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible...

The new construction given by Barton and Sudbery of the Freudenthal-Tits magic square, which includes the exceptional classical simple Lie algebras, will be interpreted and extended by using a pair of symmetric composition algebras, instead of the standard unital composition algebras.