Lie Group

Proposition

For any Lie group $G$, its tangent bundle $\begin{array}{ccc}TG={⨆}_{g\in G}{T}_{g}G& \underset{\text{Diffeomorphism}}{\overset{\Phi ,\cong }{\to }}& G\times T_{e}G\\ \pi \searrow & \circlearrowright & \swarrow p_1\\ {\pi }^{-1}(g)=T_{g}G& g\in G& p_1^{-1}(g)=T_{e}G\end{array}$ is a “bundle” isomorphism. $\Phi$ is a linear isomorphism along each fiber.

Corollary

The tangent bundle of Lie Group is a trivial bundle.