Tangent space at $SO(3)$ is isomorphic to tangent space at identity matrix

Question

I am reading a paper, the following is the part of it:

I can not quite understand the green part.

It seems that every tangent space at an element of $SO(3)$ is isomorphic to the tangent space at the identity element $I$, which is also in $SO(3)$. And we define this tangent space as the Lie algebra $so(3)$ of the Lie group $SO(3)$.

Could anyone please explain this fact or prove this?

Answer 1

If $G$ is a Lie group and $g\in G$, then the tangent space $T_gG$ at $g$ is canonically isomorphic to the tangent space $T_1G$ at $1$ via $$(dL_{g^-1})_g:T_gG\to T_1G.$$ Here $L_{g^{-1}}:G\to G$ is left translation by $g^{-1}$ i.e. $a\mapsto g^{-1}a$, and $(dL_{g^-1})_g$ is the differential of $L_{g^{-1}}$ at $g$.