A geometric characterization of the symplectic Lie algebra

Abstract

A nonzero element $x$ in a Lie algebra $\mathfrak{g}$ with Lie product $[,]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an ${\mathfrak{sl}}_2$ and any third extremal element $z$ commutes with at least one extremal element in this ${\mathfrak{sl}}_2$.