  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
$\mathfrak{sl}_2$.
