Cartesian product of diffeomorphic Lie groups

Question

Let $A,B,$ and $C$ be Lie groups. If $A$ and $B$ are diffeomorphic, are the Cartesian products $A \times C$ and $B\times C$ diffeomorphic? In other words, does $A \cong B$ imply $A \times C \cong B\times C$?

If not, are there some conditions on the groups such that the above implication holds? Many thanks.

Answer 1

This has nothing to do with groups, and true for any manifolds $A, B, C$ as above. If the diffeomorphism is $d: A\to B,$ then $D: (A, C) \to (B, C)$ by $D(a, c) = (d(a), c)$ works wonders.