  The set of exact solutions of the non-linear realisations of the G+++
Kac-Moody algebras is further analysed. Intersection rules for extremal branes
translate into orthogonality conditions on the positive real roots
characterising each brane. It is proven that all the intersecting extremal
brane solutions of the maximally oxidised theories have their algebraic
counterparts as exact solutions in the G+++ invariant theories. The proof is
extended to include the intersecting extremal brane solutions of the exotic
phases of the maximally oxidised theories.
