  An extremal element $x$ in a Lie algebra $\mathfrak{g}$ is an element for
which the space $[x, [x, \mathfrak{g}]]$ is contained in the linear span of
$x$. Long root elements in classical Lie algebras are examples of extremal
elements. Lie algebras generated by extremal elements lead to geometries with
as points the $1$-spaces generate by extremal elements an as lines the
$2$-spaces whose non-zero elements are pairwise commuting extremal elements.
  In this paper we show that the finitary special linear Lie algebras can be
characterized by their extremal geometry. Moreover, we also show that the
finitary special unitary Lie algebras can be characterized by the fact that
their geometry has no lines, but that after extending the field quadratically,
the geometry is that of a special linear Lie algebra.
