Example 3

In particular, when $B = B'$, the B-space $E\times_BE'$ is identified with the principal fibre space with group $G\times G'$ induced by $E\times E'$ over the diagonal $\Delta_B$ of $B\times B$. Endowed with this structure of principal fibre space, the B-space $E\times_BE'$ is called the fibre product of the principal fibre spaces E and $E'$.