Recovering the Lie algebra from its extremal geometry

An element x of a Lie algebra L over the field F   is extremal if [x,[x,L]]=Fx[x,[x,L]]=Fx. Under minor assumptions, it is known that, for a simple Lie algebra L  , the extremal geometry E(L)E(L) is a subspace of the projective geometry of L and either has no lines or is the root shadow space of an irreducible spherical building Δ. We prove that if Δ is of simply-laced type, then L is a quotient of a Chevalley algebra of the same type.