  We associate a root system to a finite set in a free abelian group and prove
that its irreducible subsystem is of type A, B or D. We apply this general
result to a torus manifold, where a torus manifold is a $2n$-dimensional
connected closed smooth manifold with a smooth effective action of an
$n$-dimensional compact torus having a fixed point, and show that if the torus
action extends to a smooth action of a connected compact Lie group $G$, then a
simple factor of the Lie algebra of $G$ is of type A, B or D. This gives an
alternative proof to Wiemeler's theorem. We also discuss a similar problem for
a torus manifold with an invariant stably complex structure. In this case only
type A appears.
