  Let $\Lambda$ be a simply laced root lattice and $w$ an elliptic automorphism
of $\Lambda$ of order $d$. This paper gives a construction that begins with a
central extension of the group of coinvariants $\Lambda_w$ and produces a
semisimple Lie algebra of Dykin type $\Lambda$ with an automorphism of order
$d$ lifting $w$. The input for this construction naturally arises when
considering certain families of algebraic curves. The construction generalizes
one used by Thorne to study plane quartics and one used by the author and
Thorne to study a family of genus-2 curves.
