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 epsilon(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 Delta. We prove that if Delta is of simply-laced type, then L is a quotient of a Chevalley algebra of the same type. (C) 2015 Elsevier Inc. All rights reserved.