I was able to prove that every compact Lie group is isomorphic to $$(T^n\times G_1\times \cdots \times G_m)/\Gamma$$ where the $G_i$ are compact, simply connected, simple Lie groups and $\Gamma$ is a finite subgroup of the center. However, I read somewhere that it is also true that a compact Lie group is isomorphic to $$T^n\times G_1\times \cdots \times G_m$$ where the $G_i$ are compact and simple Lie groups.
I'm sure it's a simple matter to go from the first result to the second, but I can't seem to figure it out. Any help would be greatly appreciated!
Even assuming connectedness as Moishe points out in the comments, the second claim is wrong, and
$$SO(4) \cong \left( SU(2) \times SU(2) \right) / \mathbb{Z}_2$$
is a counterexample. The relevant central copy of $\mathbb{Z}_2$ here is the diagonal one, not the left or right one, and this prevents $SO(4)$ from itself being a product of compact simple Lie groups.