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. Under minor assumptions, it is known that, for a simple Lie algebra L, the extremal geometry 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.