  For any finite graph Gamma and any field K of characteristic unequal to 2 we
construct an algebraic variety X over K whose K-points parameterise K-Lie
algebras generated by extremal elements, corresponding to the vertices of the
graph, with prescribed commutation relations, corresponding to the non-edges.
After that, we study the case where Gamma is a connected, simply laced Dynkin
diagram of finite or affine type. We prove that X is then an affine space, and
that all points in an open dense subset of X parameterise Lie algebras
isomorphic to a single fixed Lie algebra. If Gamma is of affine type, then this
fixed Lie algebra is the split finite-dimensional simple Lie algebra
corresponding to the associated finite-type Dynkin diagram. This gives a new
construction of these Lie algebras, in which they come together with
interesting degenerations, corresponding to points outside the open dense
subset. Our results may prove useful for recognising these Lie algebras.
