Root systems and symmetries of torus manifolds

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.