Remark 1

Let $(E, p)$ be a principal fibre space with base B and group G. With the notation of proposition 3, the mapping $\theta^{-1}: E\times_BE\rightarrow E\times G$ is a trivialisation of the principal fibre space pr$_1: E\times_BE\rightarrow E$. One says that $\theta^{-1}$ is the canonical trivialisation of this principal fibre space.