Let $G$ be a connected Lie group with a compact semisimple Lie algebra. It is well known that the Lie algebra then can be written as the product of compact simple Lie algebras which are classified by Dynkin diagrams. I would like to know what we can say on the level of Lie groups, i.e. is it true that the Lie group then can be written as a product of compact simple Lie groups?
No, one has to take into account coverings. The smallest counterexample is $SO(4)$, which is not such a product, but whose universal cover is a product $Spin(4) \cong SU(2) \times SU(2)$.