Suppose $A,B$ are smooth manifolds and there exists a binary operation on the product manifold $A \times B$ making it into a Lie group.
- Does this guarantee that there exist binary operations on both $A$ and $B$ making them each into Lie groups?
- If the answer is yes, can it be done in such a way that the product group $A \times B$ is equal to the original Lie group?
- If the answer to that is yes, is it necessary?
I tried briefly to find a counterexample browsing a table of Lie groups but was unsuccessful.