G+++ Invariant Formulation of Gravity and M-Theories: Exact Intersecting   Brane Solutions
  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. 
Francois Englert, Laurent Houart