Let $H_1$ and $H_2$ be two Lie groups with only finitely many connected components. Consider the two statements.
A) There is a connected Lie group $P$ such that there exist finite sheeted coverings from $P$ to both $H_1$ and $H_2$.
B) There is a connected Lie group $C$ such that there exist finite sheeted coverings from both $H_1$ and $H_2$ to $C$.
It is easy to show that (B) implies (A) since the fibred product construction gives $P$ from $C$. However, in the definition of isogeny of two Lie groups given in Chapter 3 of D. W. Morris's Introduction to Arithmetic Groups it is stated that the statements (A) and (B) are equivalent.
I am unable to see how to infer (B) from (A).
A covering of connected Lie groups is just a quotient by a discrete subgroup of the center. So (A) means that there are finite subgroups $D_1,D_2\subseteq Z(P)$ such that $P/D_1\cong H_1$ and $P/D_2\cong H_2$ (or, the connected components of the identity of $H_1$ and $H_2$, if you don't require covers to be surjective). We then see that $D_1D_2$ is another finite and hence discrete subgroup of $Z(P)$, and so $P/(D_1D_2)$ is another group covered by $P$ which has both $P/D_1\cong H_1$ and $P/D_2\cong H_2$ as finite-sheeted covers.